id	sid	tid	token	lemma	pos
ejpam-4835	1	1	european	european	PROPN
ejpam-4835	1	2	journal	journal	PROPN
ejpam-4835	1	3	of	of	ADP
ejpam-4835	1	4	pure	pure	ADJ
ejpam-4835	1	5	and	and	CCONJ
ejpam-4835	1	6	applied	apply	VERB
ejpam-4835	1	7	mathematics	mathematic	NOUN
ejpam-4835	1	8	vol	vol	NOUN
ejpam-4835	1	9	.	.	PUNCT
ejpam-4835	2	1	16	16	NUM
ejpam-4835	2	2	,	,	PUNCT
ejpam-4835	2	3	no	no	INTJ
ejpam-4835	2	4	.	.	NOUN
ejpam-4835	2	5	3	3	NUM
ejpam-4835	2	6	,	,	PUNCT
ejpam-4835	2	7	2023	2023	NUM
ejpam-4835	2	8	,	,	PUNCT
ejpam-4835	2	9	1956	1956	NUM
ejpam-4835	2	10	-	-	SYM
ejpam-4835	2	11	1969	1969	NUM
ejpam-4835	2	12	issn	issn	PROPN
ejpam-4835	2	13	1307	1307	NUM
ejpam-4835	2	14	-	-	SYM
ejpam-4835	2	15	5543	5543	NUM
ejpam-4835	2	16	–	–	PUNCT
ejpam-4835	2	17	ejpam.com	ejpam.com	X
ejpam-4835	2	18	published	publish	VERB
ejpam-4835	2	19	by	by	ADP
ejpam-4835	2	20	new	new	PROPN
ejpam-4835	2	21	york	york	PROPN
ejpam-4835	2	22	business	business	PROPN
ejpam-4835	2	23	global	global	ADJ
ejpam-4835	2	24	multigrid	multigrid	NOUN
ejpam-4835	2	25	methods	method	NOUN
ejpam-4835	2	26	for	for	ADP
ejpam-4835	2	27	the	the	DET
ejpam-4835	2	28	solution	solution	NOUN
ejpam-4835	2	29	of	of	ADP
ejpam-4835	2	30	nonlinear	nonlinear	ADJ
ejpam-4835	2	31	variational	variational	ADJ
ejpam-4835	2	32	inequalities	inequality	NOUN
ejpam-4835	2	33	nesba	nesba	PROPN
ejpam-4835	2	34	nour	nour	PROPN
ejpam-4835	2	35	el	el	PROPN
ejpam-4835	2	36	houda1	houda1	PROPN
ejpam-4835	2	37	,	,	PUNCT
ejpam-4835	2	38	beggas	beggas	NOUN
ejpam-4835	2	39	mohammed1	mohammed1	NOUN
ejpam-4835	2	40	,	,	PUNCT
ejpam-4835	2	41	belouafi	belouafi	PROPN
ejpam-4835	2	42	mohammed	mohammed	PROPN
ejpam-4835	2	43	essaid1	essaid1	PROPN
ejpam-4835	2	44	,	,	PUNCT
ejpam-4835	2	45	imtiaz	imtiaz	PROPN
ejpam-4835	2	46	ahmad2	ahmad2	PROPN
ejpam-4835	2	47	,	,	PUNCT
ejpam-4835	2	48	hijaz	hijaz	PROPN
ejpam-4835	2	49	ahmad3,4,5,∗	ahmad3,4,5,∗	PROPN
ejpam-4835	2	50	,	,	PUNCT
ejpam-4835	2	51	sameh	sameh	NOUN
ejpam-4835	2	52	askar6	askar6	NOUN
ejpam-4835	2	53	1	1	NUM
ejpam-4835	2	54	faculty	faculty	NOUN
ejpam-4835	2	55	of	of	ADP
ejpam-4835	2	56	exact	exact	ADJ
ejpam-4835	2	57	sciences	science	NOUN
ejpam-4835	2	58	,	,	PUNCT
ejpam-4835	2	59	echahid	echahid	NOUN
ejpam-4835	2	60	hamma	hamma	PROPN
ejpam-4835	2	61	lakhdar	lakhdar	PROPN
ejpam-4835	2	62	university	university	PROPN
ejpam-4835	2	63	of	of	ADP
ejpam-4835	2	64	el	el	PROPN
ejpam-4835	2	65	oued	oued	PROPN
ejpam-4835	2	66	,	,	PUNCT
ejpam-4835	2	67	operator	operator	NOUN
ejpam-4835	2	68	theory	theory	NOUN
ejpam-4835	2	69	,	,	PUNCT
ejpam-4835	2	70	pde	pde	PROPN
ejpam-4835	2	71	and	and	CCONJ
ejpam-4835	2	72	applications	application	NOUN
ejpam-4835	2	73	laboratory	laboratory	NOUN
ejpam-4835	2	74	,	,	PUNCT
ejpam-4835	2	75	algeria	algeria	PROPN
ejpam-4835	2	76	2	2	NUM
ejpam-4835	2	77	institute	institute	NOUN
ejpam-4835	2	78	of	of	ADP
ejpam-4835	2	79	informatics	informatic	NOUN
ejpam-4835	2	80	and	and	CCONJ
ejpam-4835	2	81	computing	compute	VERB
ejpam-4835	2	82	in	in	ADP
ejpam-4835	2	83	energy	energy	NOUN
ejpam-4835	2	84	(	(	PUNCT
ejpam-4835	2	85	iice	iice	NOUN
ejpam-4835	2	86	)	)	PUNCT
ejpam-4835	2	87	,	,	PUNCT
ejpam-4835	2	88	universiti	universiti	PROPN
ejpam-4835	2	89	tenaga	tenaga	PROPN
ejpam-4835	2	90	nasional	nasional	PROPN
ejpam-4835	2	91	,	,	PUNCT
ejpam-4835	2	92	kajang	kajang	PROPN
ejpam-4835	2	93	,	,	PUNCT
ejpam-4835	2	94	selangor	selangor	PROPN
ejpam-4835	2	95	,	,	PUNCT
ejpam-4835	2	96	malaysia	malaysia	PROPN
ejpam-4835	2	97	3	3	NUM
ejpam-4835	2	98	section	section	NOUN
ejpam-4835	2	99	of	of	ADP
ejpam-4835	2	100	mathematics	mathematic	NOUN
ejpam-4835	2	101	,	,	PUNCT
ejpam-4835	2	102	international	international	ADJ
ejpam-4835	2	103	telematic	telematic	ADJ
ejpam-4835	2	104	university	university	NOUN
ejpam-4835	2	105	uninettuno	uninettuno	NOUN
ejpam-4835	2	106	,	,	PUNCT
ejpam-4835	2	107	corso	corso	PROPN
ejpam-4835	2	108	vittorio	vittorio	PROPN
ejpam-4835	2	109	emanuele	emanuele	PROPN
ejpam-4835	2	110	ii	ii	PROPN
ejpam-4835	2	111	,	,	PUNCT
ejpam-4835	2	112	39,00186	39,00186	PROPN
ejpam-4835	2	113	roma	roma	PROPN
ejpam-4835	2	114	,	,	PUNCT
ejpam-4835	2	115	italy	italy	PROPN
ejpam-4835	2	116	4	4	NUM
ejpam-4835	2	117	operational	operational	ADJ
ejpam-4835	2	118	research	research	NOUN
ejpam-4835	2	119	centre	centre	NOUN
ejpam-4835	2	120	in	in	ADP
ejpam-4835	2	121	healthcare	healthcare	PROPN
ejpam-4835	2	122	,	,	PUNCT
ejpam-4835	2	123	near	near	ADP
ejpam-4835	2	124	east	east	PROPN
ejpam-4835	2	125	university	university	PROPN
ejpam-4835	2	126	,	,	PUNCT
ejpam-4835	2	127	trnc	trnc	PROPN
ejpam-4835	2	128	mersin	mersin	PROPN
ejpam-4835	2	129	10	10	NUM
ejpam-4835	2	130	,	,	PUNCT
ejpam-4835	2	131	nicosia	nicosia	NOUN
ejpam-4835	2	132	,	,	PUNCT
ejpam-4835	2	133	99138	99138	NUM
ejpam-4835	2	134	,	,	PUNCT
ejpam-4835	2	135	turkey	turkey	PROPN
ejpam-4835	2	136	5	5	NUM
ejpam-4835	2	137	department	department	NOUN
ejpam-4835	2	138	of	of	ADP
ejpam-4835	2	139	computer	computer	NOUN
ejpam-4835	2	140	science	science	NOUN
ejpam-4835	2	141	and	and	CCONJ
ejpam-4835	2	142	mathematics	mathematic	NOUN
ejpam-4835	2	143	,	,	PUNCT
ejpam-4835	2	144	lebanese	lebanese	ADJ
ejpam-4835	2	145	american	american	PROPN
ejpam-4835	2	146	university	university	PROPN
ejpam-4835	2	147	,	,	PUNCT
ejpam-4835	2	148	beirut	beirut	PROPN
ejpam-4835	2	149	,	,	PUNCT
ejpam-4835	2	150	lebanon	lebanon	PROPN
ejpam-4835	2	151	6	6	NUM
ejpam-4835	2	152	department	department	NOUN
ejpam-4835	2	153	of	of	ADP
ejpam-4835	2	154	statistics	statistic	NOUN
ejpam-4835	2	155	and	and	CCONJ
ejpam-4835	2	156	operations	operation	NOUN
ejpam-4835	2	157	research	research	NOUN
ejpam-4835	2	158	,	,	PUNCT
ejpam-4835	2	159	college	college	NOUN
ejpam-4835	2	160	of	of	ADP
ejpam-4835	2	161	science	science	NOUN
ejpam-4835	2	162	,	,	PUNCT
ejpam-4835	2	163	king	king	PROPN
ejpam-4835	2	164	saud	saud	PROPN
ejpam-4835	2	165	university	university	PROPN
ejpam-4835	2	166	,	,	PUNCT
ejpam-4835	2	167	p.o	p.o	PROPN
ejpam-4835	2	168	.	.	PROPN
ejpam-4835	2	169	box	box	PROPN
ejpam-4835	2	170	2455	2455	NUM
ejpam-4835	2	171	,	,	PUNCT
ejpam-4835	2	172	riyadh	riyadh	PROPN
ejpam-4835	2	173	11451	11451	NUM
ejpam-4835	2	174	,	,	PUNCT
ejpam-4835	2	175	saudi	saudi	PROPN
ejpam-4835	2	176	arabia	arabia	PROPN
ejpam-4835	2	177	abstract	abstract	NOUN
ejpam-4835	2	178	.	.	PUNCT
ejpam-4835	3	1	in	in	ADP
ejpam-4835	3	2	this	this	DET
ejpam-4835	3	3	research	research	NOUN
ejpam-4835	3	4	,	,	PUNCT
ejpam-4835	3	5	we	we	PRON
ejpam-4835	3	6	investigate	investigate	VERB
ejpam-4835	3	7	the	the	DET
ejpam-4835	3	8	numerical	numerical	ADJ
ejpam-4835	3	9	solution	solution	NOUN
ejpam-4835	3	10	of	of	ADP
ejpam-4835	3	11	second	second	ADJ
ejpam-4835	3	12	member	member	NOUN
ejpam-4835	3	13	problems	problem	NOUN
ejpam-4835	3	14	that	that	PRON
ejpam-4835	3	15	depend	depend	VERB
ejpam-4835	3	16	on	on	ADP
ejpam-4835	3	17	the	the	DET
ejpam-4835	3	18	solution	solution	NOUN
ejpam-4835	3	19	obtained	obtain	VERB
ejpam-4835	3	20	through	through	ADP
ejpam-4835	3	21	a	a	DET
ejpam-4835	3	22	multigrid	multigrid	NOUN
ejpam-4835	3	23	method	method	NOUN
ejpam-4835	3	24	.	.	PUNCT
ejpam-4835	4	1	specifically	specifically	ADV
ejpam-4835	4	2	,	,	PUNCT
ejpam-4835	4	3	we	we	PRON
ejpam-4835	4	4	focus	focus	VERB
ejpam-4835	4	5	on	on	ADP
ejpam-4835	4	6	the	the	DET
ejpam-4835	4	7	application	application	NOUN
ejpam-4835	4	8	of	of	ADP
ejpam-4835	4	9	multigrid	multigrid	NOUN
ejpam-4835	4	10	techniques	technique	NOUN
ejpam-4835	4	11	for	for	ADP
ejpam-4835	4	12	solving	solve	VERB
ejpam-4835	4	13	nonlinear	nonlinear	ADJ
ejpam-4835	4	14	variational	variational	ADJ
ejpam-4835	4	15	inequalities	inequality	NOUN
ejpam-4835	4	16	.	.	PUNCT
ejpam-4835	5	1	the	the	DET
ejpam-4835	5	2	main	main	ADJ
ejpam-4835	5	3	objective	objective	NOUN
ejpam-4835	5	4	is	be	AUX
ejpam-4835	5	5	to	to	PART
ejpam-4835	5	6	establish	establish	VERB
ejpam-4835	5	7	the	the	DET
ejpam-4835	5	8	uniform	uniform	ADJ
ejpam-4835	5	9	convergence	convergence	NOUN
ejpam-4835	5	10	of	of	ADP
ejpam-4835	5	11	the	the	DET
ejpam-4835	5	12	multigrid	multigrid	NOUN
ejpam-4835	5	13	algorithm	algorithm	NOUN
ejpam-4835	5	14	.	.	PUNCT
ejpam-4835	6	1	to	to	PART
ejpam-4835	6	2	achieve	achieve	VERB
ejpam-4835	6	3	this	this	PRON
ejpam-4835	6	4	,	,	PUNCT
ejpam-4835	6	5	we	we	PRON
ejpam-4835	6	6	employ	employ	VERB
ejpam-4835	6	7	elementary	elementary	ADJ
ejpam-4835	6	8	subdifferential	subdifferential	ADJ
ejpam-4835	6	9	calculus	calculus	NOUN
ejpam-4835	6	10	and	and	CCONJ
ejpam-4835	6	11	draw	draw	VERB
ejpam-4835	6	12	insights	insight	NOUN
ejpam-4835	6	13	from	from	ADP
ejpam-4835	6	14	the	the	DET
ejpam-4835	6	15	convergence	convergence	NOUN
ejpam-4835	6	16	theory	theory	NOUN
ejpam-4835	6	17	of	of	ADP
ejpam-4835	6	18	nonlinear	nonlinear	ADJ
ejpam-4835	6	19	multigrid	multigrid	NOUN
ejpam-4835	6	20	methods	method	NOUN
ejpam-4835	6	21	.	.	PUNCT
ejpam-4835	7	1	2020	2020	NUM
ejpam-4835	7	2	mathematics	mathematic	NOUN
ejpam-4835	7	3	subject	subject	NOUN
ejpam-4835	7	4	classifications	classification	NOUN
ejpam-4835	7	5	:	:	PUNCT
ejpam-4835	7	6	65m55	65m55	NUM
ejpam-4835	7	7	,	,	PUNCT
ejpam-4835	7	8	65n30	65n30	NUM
ejpam-4835	7	9	,	,	PUNCT
ejpam-4835	7	10	49n05	49n05	NUM
ejpam-4835	7	11	,	,	PUNCT
ejpam-4835	7	12	65k15	65k15	NUM
ejpam-4835	7	13	,	,	PUNCT
ejpam-4835	7	14	35j86	35j86	NUM
ejpam-4835	7	15	key	key	ADJ
ejpam-4835	7	16	words	word	NOUN
ejpam-4835	7	17	and	and	CCONJ
ejpam-4835	7	18	phrases	phrase	NOUN
ejpam-4835	7	19	:	:	PUNCT
ejpam-4835	7	20	multigrid	multigrid	NOUN
ejpam-4835	7	21	method	method	NOUN
ejpam-4835	7	22	,	,	PUNCT
ejpam-4835	7	23	nonlinear	nonlinear	ADJ
ejpam-4835	7	24	variational	variational	ADJ
ejpam-4835	7	25	inequality	inequality	NOUN
ejpam-4835	7	26	,	,	PUNCT
ejpam-4835	7	27	finite	finite	NOUN
ejpam-4835	7	28	element	element	NOUN
ejpam-4835	7	29	,	,	PUNCT
ejpam-4835	7	30	iterative	iterative	NOUN
ejpam-4835	7	31	method	method	NOUN
ejpam-4835	7	32	,	,	PUNCT
ejpam-4835	7	33	hjb	hjb	NOUN
ejpam-4835	7	34	equation	equation	NOUN
ejpam-4835	7	35	1	1	NUM
ejpam-4835	7	36	.	.	PUNCT
ejpam-4835	8	1	introduction	introduction	NOUN
ejpam-4835	8	2	the	the	DET
ejpam-4835	8	3	recent	recent	ADJ
ejpam-4835	8	4	literature	literature	NOUN
ejpam-4835	8	5	offers	offer	VERB
ejpam-4835	8	6	a	a	DET
ejpam-4835	8	7	wide	wide	ADJ
ejpam-4835	8	8	range	range	NOUN
ejpam-4835	8	9	of	of	ADP
ejpam-4835	8	10	diverse	diverse	ADJ
ejpam-4835	8	11	computational	computational	ADJ
ejpam-4835	8	12	methods	method	NOUN
ejpam-4835	8	13	[	[	X
ejpam-4835	8	14	6	6	NUM
ejpam-4835	8	15	,	,	PUNCT
ejpam-4835	8	16	9	9	NUM
ejpam-4835	8	17	,	,	PUNCT
ejpam-4835	8	18	20	20	NUM
ejpam-4835	8	19	,	,	PUNCT
ejpam-4835	8	20	21	21	NUM
ejpam-4835	8	21	,	,	PUNCT
ejpam-4835	8	22	24	24	NUM
ejpam-4835	8	23	,	,	PUNCT
ejpam-4835	8	24	29	29	NUM
ejpam-4835	8	25	,	,	PUNCT
ejpam-4835	8	26	31	31	NUM
ejpam-4835	8	27	,	,	PUNCT
ejpam-4835	8	28	32	32	NUM
ejpam-4835	8	29	,	,	PUNCT
ejpam-4835	8	30	34	34	NUM
ejpam-4835	8	31	,	,	PUNCT
ejpam-4835	8	32	35	35	NUM
ejpam-4835	8	33	]	]	PUNCT
ejpam-4835	8	34	that	that	PRON
ejpam-4835	8	35	are	be	AUX
ejpam-4835	8	36	employed	employ	VERB
ejpam-4835	8	37	to	to	PART
ejpam-4835	8	38	solve	solve	VERB
ejpam-4835	8	39	complex	complex	ADJ
ejpam-4835	8	40	real	real	ADJ
ejpam-4835	8	41	-	-	PUNCT
ejpam-4835	8	42	world	world	NOUN
ejpam-4835	8	43	problems	problem	NOUN
ejpam-4835	8	44	across	across	ADP
ejpam-4835	8	45	∗corresponding	∗corresponde	VERB
ejpam-4835	8	46	author	author	NOUN
ejpam-4835	8	47	.	.	PUNCT
ejpam-4835	9	1	doi	doi	NOUN
ejpam-4835	9	2	:	:	PUNCT
ejpam-4835	9	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4835	https://doi.org/10.29020/nybg.ejpam.v16i3.4835	NOUN
ejpam-4835	9	4	email	email	NOUN
ejpam-4835	9	5	addresses	address	NOUN
ejpam-4835	9	6	:	:	PUNCT
ejpam-4835	9	7	ahmad.hijaz@uninettuno.it	ahmad.hijaz@uninettuno.it	PROPN
ejpam-4835	9	8	hijaz.ahmad@neu.edu.tr	hijaz.ahmad@neu.edu.tr	PROPN
ejpam-4835	9	9	(	(	PUNCT
ejpam-4835	9	10	h.	h.	PROPN
ejpam-4835	9	11	ahmad	ahmad	PROPN
ejpam-4835	9	12	)	)	PUNCT
ejpam-4835	9	13	,	,	PUNCT
ejpam-4835	9	14	nour17nesba@gmail.com	nour17nesba@gmail.com	X
ejpam-4835	9	15	(	(	PUNCT
ejpam-4835	9	16	n.	n.	PROPN
ejpam-4835	9	17	n.	n.	PROPN
ejpam-4835	9	18	e.	e.	PROPN
ejpam-4835	9	19	houda	houda	PROPN
ejpam-4835	9	20	)	)	PUNCT
ejpam-4835	9	21	,	,	PUNCT
ejpam-4835	9	22	beggasmr@yahoo.fr	beggasmr@yahoo.fr	PROPN
ejpam-4835	9	23	(	(	PUNCT
ejpam-4835	9	24	b.	b.	PROPN
ejpam-4835	9	25	mohammed	mohammed	PROPN
ejpam-4835	9	26	)	)	PUNCT
ejpam-4835	9	27	,	,	PUNCT
ejpam-4835	9	28	gogoo.said@gmail.com	gogoo.said@gmail.com	X
ejpam-4835	9	29	(	(	PUNCT
ejpam-4835	9	30	b.	b.	PROPN
ejpam-4835	9	31	m.	m.	PROPN
ejpam-4835	9	32	essaid	essaid	PROPN
ejpam-4835	9	33	)	)	PUNCT
ejpam-4835	9	34	,	,	PUNCT
ejpam-4835	9	35	imtiaz.ahmad@uniten.edu.my	imtiaz.ahmad@uniten.edu.my	PROPN
ejpam-4835	9	36	(	(	PUNCT
ejpam-4835	9	37	i.	i.	PROPN
ejpam-4835	9	38	ahmad	ahmad	PROPN
ejpam-4835	9	39	)	)	PUNCT
ejpam-4835	9	40	,	,	PUNCT
ejpam-4835	9	41	saskar@ksu.edu.sa	saskar@ksu.edu.sa	PROPN
ejpam-4835	9	42	(	(	PUNCT
ejpam-4835	9	43	s.	s.	PROPN
ejpam-4835	9	44	askar	askar	PROPN
ejpam-4835	9	45	)	)	PUNCT
ejpam-4835	9	46	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4835	9	47	1956	1956	NUM
ejpam-4835	10	1	©	©	ADP
ejpam-4835	10	2	2023	2023	NUM
ejpam-4835	10	3	ejpam	ejpam	NOUN
ejpam-4835	10	4	all	all	DET
ejpam-4835	10	5	rights	right	NOUN
ejpam-4835	10	6	reserved	reserve	VERB
ejpam-4835	10	7	.	.	PUNCT
ejpam-4835	11	1	h.	h.	PROPN
ejpam-4835	11	2	ahmad	ahmad	PROPN
ejpam-4835	11	3	et	et	PROPN
ejpam-4835	11	4	al	al	PROPN
ejpam-4835	11	5	.	.	PUNCT
ejpam-4835	11	6	/	/	SYM
ejpam-4835	11	7	eur	eur	PROPN
ejpam-4835	11	8	.	.	PUNCT
ejpam-4835	12	1	j.	j.	PROPN
ejpam-4835	12	2	pure	pure	PROPN
ejpam-4835	12	3	appl	appl	PROPN
ejpam-4835	12	4	.	.	PROPN
ejpam-4835	12	5	math	math	PROPN
ejpam-4835	12	6	,	,	PUNCT
ejpam-4835	12	7	16	16	NUM
ejpam-4835	12	8	(	(	PUNCT
ejpam-4835	12	9	3	3	NUM
ejpam-4835	12	10	)	)	PUNCT
ejpam-4835	12	11	(	(	PUNCT
ejpam-4835	12	12	2023	2023	NUM
ejpam-4835	12	13	)	)	PUNCT
ejpam-4835	12	14	,	,	PUNCT
ejpam-4835	12	15	1956	1956	NUM
ejpam-4835	12	16	-	-	SYM
ejpam-4835	12	17	1969	1969	NUM
ejpam-4835	12	18	1957	1957	NUM
ejpam-4835	12	19	various	various	ADJ
ejpam-4835	12	20	fields	field	NOUN
ejpam-4835	12	21	of	of	ADP
ejpam-4835	12	22	science	science	NOUN
ejpam-4835	12	23	and	and	CCONJ
ejpam-4835	12	24	engineering	engineering	NOUN
ejpam-4835	13	1	[	[	X
ejpam-4835	13	2	1	1	NUM
ejpam-4835	13	3	,	,	PUNCT
ejpam-4835	13	4	2	2	NUM
ejpam-4835	13	5	,	,	PUNCT
ejpam-4835	13	6	5	5	NUM
ejpam-4835	13	7	,	,	PUNCT
ejpam-4835	13	8	8	8	NUM
ejpam-4835	13	9	,	,	PUNCT
ejpam-4835	13	10	23	23	NUM
ejpam-4835	13	11	,	,	PUNCT
ejpam-4835	13	12	33	33	NUM
ejpam-4835	13	13	]	]	PUNCT
ejpam-4835	13	14	.	.	PUNCT
ejpam-4835	14	1	these	these	DET
ejpam-4835	14	2	methods	method	NOUN
ejpam-4835	14	3	have	have	AUX
ejpam-4835	14	4	been	be	AUX
ejpam-4835	14	5	developed	develop	VERB
ejpam-4835	14	6	and	and	CCONJ
ejpam-4835	14	7	applied	apply	VERB
ejpam-4835	14	8	to	to	PART
ejpam-4835	14	9	tackle	tackle	VERB
ejpam-4835	14	10	challenging	challenging	ADJ
ejpam-4835	14	11	problems	problem	NOUN
ejpam-4835	14	12	and	and	CCONJ
ejpam-4835	14	13	provide	provide	VERB
ejpam-4835	14	14	effective	effective	ADJ
ejpam-4835	14	15	solutions	solution	NOUN
ejpam-4835	14	16	in	in	ADP
ejpam-4835	14	17	their	their	PRON
ejpam-4835	14	18	respective	respective	ADJ
ejpam-4835	14	19	domains	domain	NOUN
ejpam-4835	14	20	.	.	PUNCT
ejpam-4835	15	1	researchers	researcher	NOUN
ejpam-4835	15	2	have	have	AUX
ejpam-4835	15	3	explored	explore	VERB
ejpam-4835	15	4	and	and	CCONJ
ejpam-4835	15	5	utilized	utilize	VERB
ejpam-4835	15	6	these	these	DET
ejpam-4835	15	7	computational	computational	ADJ
ejpam-4835	15	8	techniques	technique	NOUN
ejpam-4835	15	9	to	to	PART
ejpam-4835	15	10	address	address	VERB
ejpam-4835	15	11	a	a	DET
ejpam-4835	15	12	multitude	multitude	NOUN
ejpam-4835	15	13	of	of	ADP
ejpam-4835	15	14	problems	problem	NOUN
ejpam-4835	15	15	,	,	PUNCT
ejpam-4835	15	16	advancing	advance	VERB
ejpam-4835	15	17	our	our	PRON
ejpam-4835	15	18	understanding	understanding	NOUN
ejpam-4835	15	19	and	and	CCONJ
ejpam-4835	15	20	facilitating	facilitate	VERB
ejpam-4835	15	21	progress	progress	NOUN
ejpam-4835	15	22	in	in	ADP
ejpam-4835	15	23	numerous	numerous	ADJ
ejpam-4835	15	24	scientific	scientific	ADJ
ejpam-4835	15	25	and	and	CCONJ
ejpam-4835	15	26	engineering	engineering	NOUN
ejpam-4835	15	27	disciplines	discipline	NOUN
ejpam-4835	15	28	[	[	X
ejpam-4835	15	29	3	3	NUM
ejpam-4835	15	30	,	,	PUNCT
ejpam-4835	15	31	4	4	NUM
ejpam-4835	15	32	,	,	PUNCT
ejpam-4835	15	33	7	7	NUM
ejpam-4835	15	34	,	,	PUNCT
ejpam-4835	15	35	27	27	NUM
ejpam-4835	15	36	,	,	PUNCT
ejpam-4835	15	37	28	28	NUM
ejpam-4835	15	38	,	,	PUNCT
ejpam-4835	15	39	30	30	NUM
ejpam-4835	15	40	]	]	PUNCT
ejpam-4835	15	41	.	.	PUNCT
ejpam-4835	16	1	the	the	DET
ejpam-4835	16	2	numerical	numerical	ADJ
ejpam-4835	16	3	methods	method	NOUN
ejpam-4835	16	4	commonly	commonly	ADV
ejpam-4835	16	5	employed	employ	VERB
ejpam-4835	16	6	to	to	PART
ejpam-4835	16	7	solve	solve	VERB
ejpam-4835	16	8	boundary	boundary	ADJ
ejpam-4835	16	9	problems	problem	NOUN
ejpam-4835	16	10	typically	typically	ADV
ejpam-4835	16	11	lead	lead	VERB
ejpam-4835	16	12	,	,	PUNCT
ejpam-4835	16	13	after	after	ADP
ejpam-4835	16	14	discretization	discretization	NOUN
ejpam-4835	16	15	,	,	PUNCT
ejpam-4835	16	16	to	to	ADP
ejpam-4835	16	17	the	the	DET
ejpam-4835	16	18	resolution	resolution	NOUN
ejpam-4835	16	19	of	of	ADP
ejpam-4835	16	20	a	a	DET
ejpam-4835	16	21	system	system	NOUN
ejpam-4835	16	22	of	of	ADP
ejpam-4835	16	23	large	large	ADJ
ejpam-4835	16	24	algebraic	algebraic	ADJ
ejpam-4835	16	25	equations	equation	NOUN
ejpam-4835	16	26	.	.	PUNCT
ejpam-4835	17	1	these	these	DET
ejpam-4835	17	2	numerical	numerical	ADJ
ejpam-4835	17	3	algorithms	algorithm	NOUN
ejpam-4835	17	4	,	,	PUNCT
ejpam-4835	17	5	including	include	VERB
ejpam-4835	17	6	iteration	iteration	NOUN
ejpam-4835	17	7	methods	method	NOUN
ejpam-4835	17	8	such	such	ADJ
ejpam-4835	17	9	as	as	ADP
ejpam-4835	17	10	jacobi	jacobi	PROPN
ejpam-4835	17	11	,	,	PUNCT
ejpam-4835	17	12	gauss	gauss	ADJ
ejpam-4835	17	13	-	-	PUNCT
ejpam-4835	17	14	seidel	seidel	PROPN
ejpam-4835	17	15	iteration	iteration	NOUN
ejpam-4835	17	16	,	,	PUNCT
ejpam-4835	17	17	and	and	CCONJ
ejpam-4835	17	18	relaxation	relaxation	NOUN
ejpam-4835	17	19	methods	method	NOUN
ejpam-4835	17	20	,	,	PUNCT
ejpam-4835	17	21	are	be	AUX
ejpam-4835	17	22	often	often	ADV
ejpam-4835	17	23	used	use	VERB
ejpam-4835	17	24	due	due	ADP
ejpam-4835	17	25	to	to	ADP
ejpam-4835	17	26	their	their	PRON
ejpam-4835	17	27	customary	customary	ADJ
ejpam-4835	17	28	nature	nature	NOUN
ejpam-4835	17	29	.	.	PUNCT
ejpam-4835	18	1	however	however	ADV
ejpam-4835	18	2	,	,	PUNCT
ejpam-4835	18	3	they	they	PRON
ejpam-4835	18	4	may	may	AUX
ejpam-4835	18	5	exhibit	exhibit	VERB
ejpam-4835	18	6	slow	slow	ADJ
ejpam-4835	18	7	convergence	convergence	NOUN
ejpam-4835	18	8	for	for	ADP
ejpam-4835	18	9	small	small	ADJ
ejpam-4835	18	10	mesh	mesh	NOUN
ejpam-4835	18	11	sizes	size	NOUN
ejpam-4835	18	12	and	and	CCONJ
ejpam-4835	18	13	complexity	complexity	NOUN
ejpam-4835	18	14	when	when	SCONJ
ejpam-4835	18	15	adapted	adapt	VERB
ejpam-4835	18	16	to	to	ADP
ejpam-4835	18	17	general	general	ADJ
ejpam-4835	18	18	elliptical	elliptical	ADJ
ejpam-4835	18	19	problems	problem	NOUN
ejpam-4835	18	20	.	.	PUNCT
ejpam-4835	19	1	in	in	ADP
ejpam-4835	19	2	contrast	contrast	NOUN
ejpam-4835	19	3	,	,	PUNCT
ejpam-4835	19	4	multigrid	multigrid	ADJ
ejpam-4835	19	5	methods	method	NOUN
ejpam-4835	19	6	offer	offer	VERB
ejpam-4835	19	7	a	a	DET
ejpam-4835	19	8	distinct	distinct	ADJ
ejpam-4835	19	9	advantage	advantage	NOUN
ejpam-4835	19	10	.	.	PUNCT
ejpam-4835	20	1	they	they	PRON
ejpam-4835	20	2	are	be	AUX
ejpam-4835	20	3	algorithms	algorithm	NOUN
ejpam-4835	20	4	with	with	ADP
ejpam-4835	20	5	a	a	DET
ejpam-4835	20	6	linear	linear	ADJ
ejpam-4835	20	7	cost	cost	NOUN
ejpam-4835	20	8	based	base	VERB
ejpam-4835	20	9	on	on	ADP
ejpam-4835	20	10	the	the	DET
ejpam-4835	20	11	number	number	NOUN
ejpam-4835	20	12	of	of	ADP
ejpam-4835	20	13	discretization	discretization	NOUN
ejpam-4835	20	14	points	point	NOUN
ejpam-4835	20	15	,	,	PUNCT
ejpam-4835	20	16	irrespective	irrespective	ADV
ejpam-4835	20	17	of	of	ADP
ejpam-4835	20	18	the	the	DET
ejpam-4835	20	19	problem	problem	NOUN
ejpam-4835	20	20	’s	’s	PART
ejpam-4835	20	21	dimension	dimension	NOUN
ejpam-4835	20	22	.	.	PUNCT
ejpam-4835	21	1	these	these	DET
ejpam-4835	21	2	methods	method	NOUN
ejpam-4835	21	3	are	be	AUX
ejpam-4835	21	4	particularly	particularly	ADV
ejpam-4835	21	5	efficient	efficient	ADJ
ejpam-4835	21	6	in	in	ADP
ejpam-4835	21	7	solving	solve	VERB
ejpam-4835	21	8	both	both	CCONJ
ejpam-4835	21	9	linear	linear	ADJ
ejpam-4835	21	10	and	and	CCONJ
ejpam-4835	21	11	nonlinear	nonlinear	ADJ
ejpam-4835	21	12	partial	partial	ADJ
ejpam-4835	21	13	differential	differential	NOUN
ejpam-4835	21	14	equations	equation	NOUN
ejpam-4835	21	15	(	(	PUNCT
ejpam-4835	21	16	pdes	pde	NOUN
ejpam-4835	21	17	)	)	PUNCT
ejpam-4835	21	18	and	and	CCONJ
ejpam-4835	21	19	linear	linear	ADJ
ejpam-4835	21	20	variational	variational	ADJ
ejpam-4835	21	21	inequalities	inequality	NOUN
ejpam-4835	21	22	(	(	PUNCT
ejpam-4835	21	23	vis	vis	X
ejpam-4835	21	24	)	)	PUNCT
ejpam-4835	21	25	[	[	X
ejpam-4835	21	26	16	16	NUM
ejpam-4835	21	27	,	,	PUNCT
ejpam-4835	21	28	19	19	NUM
ejpam-4835	21	29	]	]	PUNCT
ejpam-4835	21	30	.	.	PUNCT
ejpam-4835	22	1	their	their	PRON
ejpam-4835	22	2	linear	linear	ADJ
ejpam-4835	22	3	complexity	complexity	NOUN
ejpam-4835	22	4	makes	make	VERB
ejpam-4835	22	5	them	they	PRON
ejpam-4835	22	6	a	a	DET
ejpam-4835	22	7	powerful	powerful	ADJ
ejpam-4835	22	8	tool	tool	NOUN
ejpam-4835	22	9	for	for	ADP
ejpam-4835	22	10	large	large	ADJ
ejpam-4835	22	11	-	-	PUNCT
ejpam-4835	22	12	scale	scale	NOUN
ejpam-4835	22	13	problems	problem	NOUN
ejpam-4835	22	14	,	,	PUNCT
ejpam-4835	22	15	significantly	significantly	ADV
ejpam-4835	22	16	reducing	reduce	VERB
ejpam-4835	22	17	computational	computational	ADJ
ejpam-4835	22	18	efforts	effort	NOUN
ejpam-4835	22	19	while	while	SCONJ
ejpam-4835	22	20	providing	provide	VERB
ejpam-4835	22	21	accurate	accurate	ADJ
ejpam-4835	22	22	solutions	solution	NOUN
ejpam-4835	22	23	.	.	PUNCT
ejpam-4835	23	1	multigrid	multigrid	NOUN
ejpam-4835	23	2	techniques	technique	NOUN
ejpam-4835	23	3	are	be	AUX
ejpam-4835	23	4	widely	widely	ADV
ejpam-4835	23	5	acknowledged	acknowledge	VERB
ejpam-4835	23	6	as	as	ADP
ejpam-4835	23	7	rapid	rapid	ADJ
ejpam-4835	23	8	methods	method	NOUN
ejpam-4835	23	9	for	for	ADP
ejpam-4835	23	10	solving	solve	VERB
ejpam-4835	23	11	various	various	ADJ
ejpam-4835	23	12	types	type	NOUN
ejpam-4835	23	13	of	of	ADP
ejpam-4835	23	14	variational	variational	ADJ
ejpam-4835	23	15	equations	equation	NOUN
ejpam-4835	23	16	and	and	CCONJ
ejpam-4835	23	17	inequalities	inequality	NOUN
ejpam-4835	23	18	[	[	X
ejpam-4835	23	19	22	22	NUM
ejpam-4835	23	20	]	]	PUNCT
ejpam-4835	23	21	,	,	PUNCT
ejpam-4835	23	22	especially	especially	ADV
ejpam-4835	23	23	for	for	ADP
ejpam-4835	23	24	elliptic	elliptic	ADJ
ejpam-4835	23	25	problems	problem	NOUN
ejpam-4835	23	26	with	with	ADP
ejpam-4835	23	27	discretization	discretization	NOUN
ejpam-4835	23	28	leading	lead	VERB
ejpam-4835	23	29	to	to	ADP
ejpam-4835	23	30	an	an	DET
ejpam-4835	23	31	m	m	NOUN
ejpam-4835	23	32	-	-	NOUN
ejpam-4835	23	33	matrix	matrix	NOUN
ejpam-4835	23	34	[	[	X
ejpam-4835	23	35	15	15	NUM
ejpam-4835	23	36	]	]	PUNCT
ejpam-4835	23	37	.	.	PUNCT
ejpam-4835	24	1	in	in	ADP
ejpam-4835	24	2	the	the	DET
ejpam-4835	24	3	second	second	ADJ
ejpam-4835	24	4	section	section	NOUN
ejpam-4835	24	5	,	,	PUNCT
ejpam-4835	24	6	we	we	PRON
ejpam-4835	24	7	present	present	VERB
ejpam-4835	24	8	an	an	DET
ejpam-4835	24	9	overview	overview	NOUN
ejpam-4835	24	10	of	of	ADP
ejpam-4835	24	11	nonlinear	nonlinear	ADJ
ejpam-4835	24	12	variational	variational	ADJ
ejpam-4835	24	13	inequality	inequality	NOUN
ejpam-4835	24	14	(	(	PUNCT
ejpam-4835	24	15	vi	vi	NOUN
ejpam-4835	24	16	)	)	PUNCT
ejpam-4835	24	17	problems	problem	NOUN
ejpam-4835	24	18	and	and	CCONJ
ejpam-4835	24	19	their	their	PRON
ejpam-4835	24	20	discretization	discretization	NOUN
ejpam-4835	24	21	using	use	VERB
ejpam-4835	24	22	a	a	DET
ejpam-4835	24	23	conforming	conform	VERB
ejpam-4835	24	24	finite	finite	NOUN
ejpam-4835	24	25	element	element	NOUN
ejpam-4835	24	26	method	method	NOUN
ejpam-4835	24	27	p1	p1	NOUN
ejpam-4835	24	28	[	[	X
ejpam-4835	24	29	13	13	NUM
ejpam-4835	24	30	]	]	PUNCT
ejpam-4835	24	31	.	.	PUNCT
ejpam-4835	25	1	additionally	additionally	ADV
ejpam-4835	25	2	,	,	PUNCT
ejpam-4835	25	3	we	we	PRON
ejpam-4835	25	4	introduce	introduce	VERB
ejpam-4835	25	5	an	an	DET
ejpam-4835	25	6	algorithm	algorithm	NOUN
ejpam-4835	25	7	that	that	PRON
ejpam-4835	25	8	formulates	formulate	VERB
ejpam-4835	25	9	the	the	DET
ejpam-4835	25	10	vi	vi	NOUN
ejpam-4835	25	11	as	as	ADP
ejpam-4835	25	12	stationary	stationary	PROPN
ejpam-4835	25	13	hamilton	hamilton	PROPN
ejpam-4835	25	14	-	-	PUNCT
ejpam-4835	25	15	jacobibellman	jacobibellman	NOUN
ejpam-4835	25	16	equations	equation	NOUN
ejpam-4835	25	17	,	,	PUNCT
ejpam-4835	25	18	drawing	draw	VERB
ejpam-4835	25	19	inspiration	inspiration	NOUN
ejpam-4835	25	20	from	from	ADP
ejpam-4835	25	21	the	the	DET
ejpam-4835	25	22	hoppe	hoppe	PROPN
ejpam-4835	25	23	multigrid	multigrid	PROPN
ejpam-4835	25	24	method	method	NOUN
ejpam-4835	25	25	[	[	X
ejpam-4835	25	26	18	18	NUM
ejpam-4835	25	27	,	,	PUNCT
ejpam-4835	25	28	25	25	NUM
ejpam-4835	25	29	]	]	PUNCT
ejpam-4835	25	30	.	.	PUNCT
ejpam-4835	26	1	we	we	PRON
ejpam-4835	26	2	refer	refer	VERB
ejpam-4835	26	3	to	to	ADP
ejpam-4835	26	4	this	this	DET
ejpam-4835	26	5	algorithm	algorithm	NOUN
ejpam-4835	26	6	as	as	ADP
ejpam-4835	26	7	the	the	DET
ejpam-4835	26	8	multigrid	multigrid	NOUN
ejpam-4835	26	9	hierarchy	hierarchy	NOUN
ejpam-4835	26	10	jacobi	jacobi	PROPN
ejpam-4835	26	11	(	(	PUNCT
ejpam-4835	26	12	mghjb	mghjb	PROPN
ejpam-4835	26	13	)	)	PUNCT
ejpam-4835	26	14	and	and	CCONJ
ejpam-4835	26	15	provide	provide	VERB
ejpam-4835	26	16	the	the	DET
ejpam-4835	26	17	iteration	iteration	NOUN
ejpam-4835	26	18	matrices	matrix	NOUN
ejpam-4835	26	19	associated	associate	VERB
ejpam-4835	26	20	with	with	ADP
ejpam-4835	26	21	the	the	DET
ejpam-4835	26	22	algorithm	algorithm	NOUN
ejpam-4835	26	23	.	.	PUNCT
ejpam-4835	27	1	firstly	firstly	ADV
ejpam-4835	27	2	,	,	PUNCT
ejpam-4835	27	3	we	we	PRON
ejpam-4835	27	4	present	present	VERB
ejpam-4835	27	5	the	the	DET
ejpam-4835	27	6	original	original	ADJ
ejpam-4835	27	7	results	result	NOUN
ejpam-4835	27	8	concerning	concern	VERB
ejpam-4835	27	9	the	the	DET
ejpam-4835	27	10	approximation	approximation	NOUN
ejpam-4835	27	11	and	and	CCONJ
ejpam-4835	27	12	smoothing	smooth	VERB
ejpam-4835	27	13	properties	property	NOUN
ejpam-4835	27	14	in	in	ADP
ejpam-4835	27	15	the	the	DET
ejpam-4835	27	16	l∞	l∞	NOUN
ejpam-4835	27	17	norm	norm	NOUN
ejpam-4835	27	18	.	.	PUNCT
ejpam-4835	28	1	subsequently	subsequently	ADV
ejpam-4835	28	2	,	,	PUNCT
ejpam-4835	28	3	we	we	PRON
ejpam-4835	28	4	demonstrate	demonstrate	VERB
ejpam-4835	28	5	the	the	DET
ejpam-4835	28	6	uniform	uniform	ADJ
ejpam-4835	28	7	convergence	convergence	NOUN
ejpam-4835	28	8	of	of	ADP
ejpam-4835	28	9	the	the	DET
ejpam-4835	28	10	mghjb	mghjb	NOUN
ejpam-4835	28	11	algorithm	algorithm	NOUN
ejpam-4835	28	12	.	.	PUNCT
ejpam-4835	29	1	finally	finally	ADV
ejpam-4835	29	2	,	,	PUNCT
ejpam-4835	29	3	we	we	PRON
ejpam-4835	29	4	apply	apply	VERB
ejpam-4835	29	5	the	the	DET
ejpam-4835	29	6	numerical	numerical	ADJ
ejpam-4835	29	7	methods	method	NOUN
ejpam-4835	29	8	to	to	ADP
ejpam-4835	29	9	a	a	DET
ejpam-4835	29	10	specific	specific	ADJ
ejpam-4835	29	11	application	application	NOUN
ejpam-4835	29	12	where	where	SCONJ
ejpam-4835	29	13	the	the	DET
ejpam-4835	29	14	operator	operator	NOUN
ejpam-4835	29	15	is	be	AUX
ejpam-4835	29	16	linear	linear	ADJ
ejpam-4835	29	17	,	,	PUNCT
ejpam-4835	29	18	and	and	CCONJ
ejpam-4835	29	19	the	the	DET
ejpam-4835	29	20	second	second	ADJ
ejpam-4835	29	21	member	member	NOUN
ejpam-4835	29	22	is	be	AUX
ejpam-4835	29	23	nonlinear	nonlinear	ADJ
ejpam-4835	29	24	,	,	PUNCT
ejpam-4835	29	25	dependent	dependent	ADJ
ejpam-4835	29	26	on	on	ADP
ejpam-4835	29	27	the	the	DET
ejpam-4835	29	28	solution	solution	NOUN
ejpam-4835	29	29	.	.	PUNCT
ejpam-4835	30	1	in	in	ADP
ejpam-4835	30	2	this	this	DET
ejpam-4835	30	3	context	context	NOUN
ejpam-4835	30	4	,	,	PUNCT
ejpam-4835	30	5	we	we	PRON
ejpam-4835	30	6	implement	implement	VERB
ejpam-4835	30	7	the	the	DET
ejpam-4835	30	8	gauss	gauss	ADJ
ejpam-4835	30	9	-	-	PUNCT
ejpam-4835	30	10	seidel	seidel	PROPN
ejpam-4835	30	11	method	method	NOUN
ejpam-4835	30	12	and	and	CCONJ
ejpam-4835	30	13	the	the	DET
ejpam-4835	30	14	multigrid	multigrid	PROPN
ejpam-4835	30	15	methods	method	NOUN
ejpam-4835	30	16	v	v	ADP
ejpam-4835	30	17	and	and	CCONJ
ejpam-4835	30	18	w	w	NOUN
ejpam-4835	30	19	-	-	PUNCT
ejpam-4835	30	20	cycle	cycle	NOUN
ejpam-4835	30	21	.	.	PUNCT
ejpam-4835	31	1	the	the	DET
ejpam-4835	31	2	numerical	numerical	ADJ
ejpam-4835	31	3	experiments	experiment	NOUN
ejpam-4835	31	4	aim	aim	VERB
ejpam-4835	31	5	to	to	PART
ejpam-4835	31	6	assess	assess	VERB
ejpam-4835	31	7	the	the	DET
ejpam-4835	31	8	effectiveness	effectiveness	NOUN
ejpam-4835	31	9	and	and	CCONJ
ejpam-4835	31	10	performance	performance	NOUN
ejpam-4835	31	11	of	of	ADP
ejpam-4835	31	12	these	these	DET
ejpam-4835	31	13	methods	method	NOUN
ejpam-4835	31	14	in	in	ADP
ejpam-4835	31	15	solving	solve	VERB
ejpam-4835	31	16	the	the	DET
ejpam-4835	31	17	given	give	VERB
ejpam-4835	31	18	problem	problem	NOUN
ejpam-4835	31	19	.	.	PUNCT
ejpam-4835	32	1	2	2	X
ejpam-4835	32	2	.	.	X
ejpam-4835	32	3	multigrid	multigrid	PROPN
ejpam-4835	32	4	method	method	PROPN
ejpam-4835	32	5	2.1	2.1	NUM
ejpam-4835	32	6	.	.	PUNCT
ejpam-4835	33	1	symbols	symbol	NOUN
ejpam-4835	33	2	and	and	CCONJ
ejpam-4835	33	3	assumptions	assumption	NOUN
ejpam-4835	33	4	let	let	VERB
ejpam-4835	33	5	ω	ω	NOUN
ejpam-4835	33	6	be	be	AUX
ejpam-4835	33	7	an	an	DET
ejpam-4835	33	8	open	open	ADJ
ejpam-4835	33	9	set	set	NOUN
ejpam-4835	33	10	in	in	ADP
ejpam-4835	33	11	rn	rn	PROPN
ejpam-4835	33	12	with	with	ADP
ejpam-4835	33	13	sufficiently	sufficiently	ADV
ejpam-4835	33	14	regular	regular	ADJ
ejpam-4835	33	15	bounds	bound	NOUN
ejpam-4835	33	16	∂ω	∂ω	PROPN
ejpam-4835	33	17	.	.	PUNCT
ejpam-4835	34	1	for	for	ADP
ejpam-4835	34	2	u	u	NOUN
ejpam-4835	34	3	,	,	PUNCT
ejpam-4835	34	4	v	v	AUX
ejpam-4835	34	5	∈	∈	PROPN
ejpam-4835	34	6	h1(ω	h1(ω	PROPN
ejpam-4835	34	7	)	)	PUNCT
ejpam-4835	34	8	,	,	PUNCT
ejpam-4835	34	9	we	we	PRON
ejpam-4835	34	10	define	define	VERB
ejpam-4835	34	11	second	second	ADJ
ejpam-4835	34	12	-	-	PUNCT
ejpam-4835	34	13	order	order	NOUN
ejpam-4835	34	14	operators	operator	NOUN
ejpam-4835	34	15	a	a	DET
ejpam-4835	34	16	=	=	SYM
ejpam-4835	34	17	∑	∑	PROPN
ejpam-4835	34	18	1≤j	1≤j	NOUN
ejpam-4835	34	19	,	,	PUNCT
ejpam-4835	34	20	k≤n	k≤n	PROPN
ejpam-4835	34	21	ajk(x	ajk(x	PROPN
ejpam-4835	34	22	)	)	PUNCT
ejpam-4835	34	23	∂2	∂2	NOUN
ejpam-4835	34	24	∂xj∂xk	∂xj∂xk	NOUN
ejpam-4835	34	25	+	+	CCONJ
ejpam-4835	34	26	n∑	n∑	PROPN
ejpam-4835	34	27	k=1	k=1	PROPN
ejpam-4835	34	28	bk(x	bk(x	PROPN
ejpam-4835	34	29	)	)	PUNCT
ejpam-4835	34	30	∂	∂	NUM
ejpam-4835	34	31	∂xk	∂xk	PROPN
ejpam-4835	34	32	+	+	CCONJ
ejpam-4835	34	33	a0(x	a0(x	NOUN
ejpam-4835	34	34	)	)	PUNCT
ejpam-4835	34	35	.	.	PUNCT
ejpam-4835	35	1	h.	h.	PROPN
ejpam-4835	35	2	ahmad	ahmad	PROPN
ejpam-4835	35	3	et	et	PROPN
ejpam-4835	35	4	al	al	PROPN
ejpam-4835	35	5	.	.	PUNCT
ejpam-4835	35	6	/	/	SYM
ejpam-4835	35	7	eur	eur	PROPN
ejpam-4835	35	8	.	.	PUNCT
ejpam-4835	36	1	j.	j.	PROPN
ejpam-4835	36	2	pure	pure	PROPN
ejpam-4835	36	3	appl	appl	PROPN
ejpam-4835	36	4	.	.	PROPN
ejpam-4835	36	5	math	math	PROPN
ejpam-4835	36	6	,	,	PUNCT
ejpam-4835	36	7	16	16	NUM
ejpam-4835	36	8	(	(	PUNCT
ejpam-4835	36	9	3	3	NUM
ejpam-4835	36	10	)	)	PUNCT
ejpam-4835	36	11	(	(	PUNCT
ejpam-4835	36	12	2023	2023	NUM
ejpam-4835	36	13	)	)	PUNCT
ejpam-4835	36	14	,	,	PUNCT
ejpam-4835	36	15	1956	1956	NUM
ejpam-4835	36	16	-	-	SYM
ejpam-4835	36	17	1969	1969	NUM
ejpam-4835	36	18	1958	1958	NUM
ejpam-4835	36	19	the	the	DET
ejpam-4835	36	20	coefficients	coefficient	NOUN
ejpam-4835	36	21	ajk(x	ajk(x	NOUN
ejpam-4835	36	22	)	)	PUNCT
ejpam-4835	36	23	,	,	PUNCT
ejpam-4835	36	24	bk(x	bk(x	NOUN
ejpam-4835	36	25	)	)	PUNCT
ejpam-4835	36	26	,	,	PUNCT
ejpam-4835	36	27	and	and	CCONJ
ejpam-4835	36	28	b0(x	b0(x	NOUN
ejpam-4835	36	29	)	)	PUNCT
ejpam-4835	36	30	are	be	AUX
ejpam-4835	36	31	required	require	VERB
ejpam-4835	36	32	to	to	PART
ejpam-4835	36	33	possess	possess	VERB
ejpam-4835	36	34	sufficient	sufficient	ADJ
ejpam-4835	36	35	regularity	regularity	NOUN
ejpam-4835	36	36	,	,	PUNCT
ejpam-4835	36	37	ensuring	ensure	VERB
ejpam-4835	36	38	:	:	PUNCT
ejpam-4835	36	39	akj(x	akj(x	X
ejpam-4835	36	40	)	)	PUNCT
ejpam-4835	36	41	=	=	SYM
ejpam-4835	36	42	ajk(x	ajk(x	PROPN
ejpam-4835	36	43	)	)	PUNCT
ejpam-4835	36	44	;	;	PUNCT
ejpam-4835	36	45	b0(x	b0(x	X
ejpam-4835	36	46	)	)	PUNCT
ejpam-4835	36	47	≧	≧	X
ejpam-4835	37	1	β	β	X
ejpam-4835	37	2	>	>	X
ejpam-4835	37	3	0	0	PROPN
ejpam-4835	37	4	,	,	PUNCT
ejpam-4835	37	5	x	x	PUNCT
ejpam-4835	37	6	∈	∈	PROPN
ejpam-4835	37	7	ω,∑	ω,∑	PROPN
ejpam-4835	37	8	1≤j	1≤j	NOUN
ejpam-4835	37	9	,	,	PUNCT
ejpam-4835	37	10	k≤n	k≤n	PROPN
ejpam-4835	38	1	akj(x)ξkξj	akj(x)ξkξj	INTJ
ejpam-4835	38	2	≧	≧	SYM
ejpam-4835	38	3	α|ξ|2	α|ξ|2	NOUN
ejpam-4835	38	4	;	;	PUNCT
ejpam-4835	38	5	(	(	PUNCT
ejpam-4835	38	6	x	x	PUNCT
ejpam-4835	38	7	∈	∈	PROPN
ejpam-4835	38	8	ω̄	ω̄	NOUN
ejpam-4835	38	9	,	,	PUNCT
ejpam-4835	38	10	ξ	ξ	PROPN
ejpam-4835	38	11	∈	∈	PROPN
ejpam-4835	38	12	rn	rn	PROPN
ejpam-4835	38	13	,	,	PUNCT
ejpam-4835	38	14	α	α	PROPN
ejpam-4835	38	15	>	>	X
ejpam-4835	38	16	0	0	NUM
ejpam-4835	38	17	)	)	PUNCT
ejpam-4835	38	18	.	.	PUNCT
ejpam-4835	39	1	we	we	PRON
ejpam-4835	39	2	also	also	ADV
ejpam-4835	39	3	define	define	VERB
ejpam-4835	39	4	the	the	DET
ejpam-4835	39	5	associated	associated	ADJ
ejpam-4835	39	6	coercive	coercive	ADJ
ejpam-4835	39	7	continuous	continuous	ADJ
ejpam-4835	39	8	bilinear	bilinear	NOUN
ejpam-4835	39	9	form	form	NOUN
ejpam-4835	39	10	:	:	PUNCT
ejpam-4835	39	11	a(u	a(u	NOUN
ejpam-4835	39	12	,	,	PUNCT
ejpam-4835	39	13	v	v	NOUN
ejpam-4835	39	14	)	)	PUNCT
ejpam-4835	39	15	=	=	SYM
ejpam-4835	40	1	∫	∫	PROPN
ejpam-4835	40	2	ω	ω	X
ejpam-4835	40	3			PROPN
ejpam-4835	40	4	∑	∑	PROPN
ejpam-4835	40	5	1≤j	1≤j	PROPN
ejpam-4835	40	6	,	,	PUNCT
ejpam-4835	40	7	k≤n	k≤n	PROPN
ejpam-4835	40	8	ajk	ajk	PROPN
ejpam-4835	40	9	∂u	∂u	PROPN
ejpam-4835	40	10	∂xj	∂xj	PROPN
ejpam-4835	40	11	∂v	∂v	PROPN
ejpam-4835	40	12	∂xk	∂xk	PROPN
ejpam-4835	41	1	+	+	PROPN
ejpam-4835	41	2	n∑	n∑	INTJ
ejpam-4835	41	3	k=1	k=1	VERB
ejpam-4835	41	4	bk	bk	NOUN
ejpam-4835	41	5	∂u	∂u	PROPN
ejpam-4835	41	6	∂xk	∂xk	PROPN
ejpam-4835	41	7	v	v	PROPN
ejpam-4835	41	8	+	+	CCONJ
ejpam-4835	41	9	a0(x)uv	a0(x)uv	VERB
ejpam-4835	41	10			PROPN
ejpam-4835	41	11	dx	dx	PROPN
ejpam-4835	41	12	.	.	PUNCT
ejpam-4835	42	1	in	in	ADP
ejpam-4835	42	2	addition	addition	NOUN
ejpam-4835	42	3	,	,	PUNCT
ejpam-4835	42	4	we	we	PRON
ejpam-4835	42	5	consider	consider	VERB
ejpam-4835	42	6	the	the	DET
ejpam-4835	42	7	second	second	ADJ
ejpam-4835	42	8	member	member	NOUN
ejpam-4835	42	9	f	f	PROPN
ejpam-4835	42	10	is	be	AUX
ejpam-4835	42	11	l	l	NOUN
ejpam-4835	42	12	-	-	ADJ
ejpam-4835	42	13	lipschitzian	lipschitzian	ADJ
ejpam-4835	42	14	such	such	ADJ
ejpam-4835	42	15	that	that	SCONJ
ejpam-4835	42	16	:	:	PUNCT
ejpam-4835	42	17	f(u	f(u	PROPN
ejpam-4835	42	18	)	)	PUNCT
ejpam-4835	42	19	∈	∈	PROPN
ejpam-4835	42	20	l∞(ω	l∞(ω	NOUN
ejpam-4835	42	21	)	)	PUNCT
ejpam-4835	42	22	;	;	PUNCT
ejpam-4835	43	1	f	f	PROPN
ejpam-4835	43	2	≥	≥	NOUN
ejpam-4835	43	3	0	0	NUM
ejpam-4835	43	4	and	and	CCONJ
ejpam-4835	43	5	l	l	NOUN
ejpam-4835	43	6	β	β	X
ejpam-4835	43	7	<	<	X
ejpam-4835	43	8	1	1	NUM
ejpam-4835	43	9	.	.	PUNCT
ejpam-4835	43	10	an	an	DET
ejpam-4835	43	11	obstacle	obstacle	NOUN
ejpam-4835	43	12	ψ	ψ	ADP
ejpam-4835	43	13	∈w	∈w	PROPN
ejpam-4835	43	14	2,∞	2,∞	NUM
ejpam-4835	43	15	,	,	PUNCT
ejpam-4835	43	16	such	such	ADJ
ejpam-4835	43	17	that	that	SCONJ
ejpam-4835	43	18	ψ	ψ	X
ejpam-4835	43	19	≥	≥	NOUN
ejpam-4835	43	20	0	0	NUM
ejpam-4835	43	21	.	.	PUNCT
ejpam-4835	43	22	2.2	2.2	NUM
ejpam-4835	43	23	.	.	PUNCT
ejpam-4835	43	24	continuous	continuous	ADJ
ejpam-4835	43	25	problem	problem	NOUN
ejpam-4835	43	26	the	the	DET
ejpam-4835	43	27	objective	objective	NOUN
ejpam-4835	43	28	is	be	AUX
ejpam-4835	43	29	to	to	PART
ejpam-4835	43	30	find	find	VERB
ejpam-4835	43	31	the	the	DET
ejpam-4835	43	32	solution	solution	NOUN
ejpam-4835	43	33	u	u	NOUN
ejpam-4835	43	34	for	for	ADP
ejpam-4835	43	35	the	the	DET
ejpam-4835	43	36	given	give	VERB
ejpam-4835	43	37	variational	variational	ADJ
ejpam-4835	43	38	inequalities	inequality	NOUN
ejpam-4835	43	39	:	:	PUNCT
ejpam-4835	43	40	find	find	VERB
ejpam-4835	43	41	u	u	NOUN
ejpam-4835	43	42	as	as	ADP
ejpam-4835	43	43	the	the	DET
ejpam-4835	43	44	solution	solution	NOUN
ejpam-4835	43	45	to	to	PART
ejpam-4835	43	46	:	:	PUNCT
ejpam-4835	43	47	{	{	PUNCT
ejpam-4835	43	48	a(u	a(u	X
ejpam-4835	43	49	,	,	PUNCT
ejpam-4835	43	50	v	v	ADP
ejpam-4835	43	51	−	−	PROPN
ejpam-4835	43	52	u	u	NOUN
ejpam-4835	43	53	)	)	PUNCT
ejpam-4835	43	54	≥	≥	PROPN
ejpam-4835	43	55	(	(	PUNCT
ejpam-4835	43	56	f(u	f(u	PROPN
ejpam-4835	43	57	)	)	PUNCT
ejpam-4835	43	58	,	,	PUNCT
ejpam-4835	43	59	v	v	ADP
ejpam-4835	43	60	−	−	PROPN
ejpam-4835	43	61	u	u	NOUN
ejpam-4835	43	62	)	)	PUNCT
ejpam-4835	43	63	,	,	PUNCT
ejpam-4835	43	64	∀v	∀v	PROPN
ejpam-4835	43	65	∈	∈	PROPN
ejpam-4835	43	66	h1(ω	h1(ω	NOUN
ejpam-4835	43	67	)	)	PUNCT
ejpam-4835	43	68	u	u	NOUN
ejpam-4835	43	69	≤	≤	NOUN
ejpam-4835	43	70	ψ	ψ	NOUN
ejpam-4835	43	71	;	;	PUNCT
ejpam-4835	43	72	v	v	NUM
ejpam-4835	43	73	≤	≤	NUM
ejpam-4835	43	74	ψ	ψ	X
ejpam-4835	43	75	(	(	PUNCT
ejpam-4835	43	76	2.3	2.3	NUM
ejpam-4835	43	77	)	)	PUNCT
ejpam-4835	43	78	it	it	PRON
ejpam-4835	43	79	is	be	AUX
ejpam-4835	43	80	established	establish	VERB
ejpam-4835	43	81	that	that	SCONJ
ejpam-4835	43	82	this	this	DET
ejpam-4835	43	83	problem	problem	NOUN
ejpam-4835	43	84	possesses	possess	VERB
ejpam-4835	43	85	a	a	DET
ejpam-4835	43	86	unique	unique	ADJ
ejpam-4835	43	87	solution	solution	NOUN
ejpam-4835	43	88	,	,	PUNCT
ejpam-4835	43	89	as	as	SCONJ
ejpam-4835	43	90	proven	prove	VERB
ejpam-4835	43	91	by	by	ADP
ejpam-4835	43	92	the	the	DET
ejpam-4835	43	93	fixed	fix	VERB
ejpam-4835	43	94	point	point	NOUN
ejpam-4835	43	95	theorem	theorem	VERB
ejpam-4835	43	96	under	under	ADP
ejpam-4835	43	97	the	the	DET
ejpam-4835	43	98	preceding	precede	VERB
ejpam-4835	43	99	assumption	assumption	NOUN
ejpam-4835	43	100	(	(	PUNCT
ejpam-4835	43	101	see	see	VERB
ejpam-4835	43	102	[	[	X
ejpam-4835	43	103	10	10	NUM
ejpam-4835	43	104	]	]	NUM
ejpam-4835	43	105	)	)	PUNCT
ejpam-4835	43	106	.	.	PUNCT
ejpam-4835	44	1	2.3	2.3	NUM
ejpam-4835	44	2	.	.	PUNCT
ejpam-4835	44	3	discretize	discretize	VERB
ejpam-4835	44	4	to	to	PART
ejpam-4835	44	5	build	build	VERB
ejpam-4835	44	6	a	a	DET
ejpam-4835	44	7	multigrid	multigrid	NOUN
ejpam-4835	44	8	loop	loop	NOUN
ejpam-4835	44	9	,	,	PUNCT
ejpam-4835	44	10	we	we	PRON
ejpam-4835	44	11	consider	consider	VERB
ejpam-4835	44	12	a	a	DET
ejpam-4835	44	13	sequence	sequence	NOUN
ejpam-4835	44	14	of	of	ADP
ejpam-4835	44	15	discretization	discretization	NOUN
ejpam-4835	44	16	steps	step	NOUN
ejpam-4835	44	17	,	,	PUNCT
ejpam-4835	44	18	denoted	denote	VERB
ejpam-4835	44	19	as	as	ADP
ejpam-4835	44	20	0	0	NUM
ejpam-4835	44	21	<	<	X
ejpam-4835	44	22	hk+1	hk+1	X
ejpam-4835	44	23	<	<	X
ejpam-4835	44	24	hk	hk	X
ejpam-4835	44	25	<	<	X
ejpam-4835	44	26	1	1	NUM
ejpam-4835	44	27	,	,	PUNCT
ejpam-4835	44	28	such	such	ADJ
ejpam-4835	44	29	that	that	SCONJ
ejpam-4835	44	30	the	the	DET
ejpam-4835	44	31	grids	grid	NOUN
ejpam-4835	44	32	are	be	AUX
ejpam-4835	44	33	nested	nest	VERB
ejpam-4835	44	34	hk+1	hk+1	NOUN
ejpam-4835	44	35	=	=	SYM
ejpam-4835	44	36	hk	hk	PROPN
ejpam-4835	44	37	2	2	NUM
ejpam-4835	44	38	.	.	PUNCT
ejpam-4835	45	1	next	next	ADV
ejpam-4835	45	2	,	,	PUNCT
ejpam-4835	45	3	we	we	PRON
ejpam-4835	45	4	define	define	VERB
ejpam-4835	45	5	ωk	ωk	ADP
ejpam-4835	45	6	=	=	NOUN
ejpam-4835	45	7	ωhk	ωhk	NOUN
ejpam-4835	45	8	,	,	PUNCT
ejpam-4835	45	9	vk	vk	VERB
ejpam-4835	45	10	=	=	PUNCT
ejpam-4835	45	11	vhk	vhk	NOUN
ejpam-4835	45	12	,	,	PUNCT
ejpam-4835	45	13	and	and	CCONJ
ejpam-4835	45	14	ak	ak	PROPN
ejpam-4835	45	15	=	=	PROPN
ejpam-4835	45	16	ahk	ahk	NOUN
ejpam-4835	45	17	,	,	PUNCT
ejpam-4835	45	18	and	and	CCONJ
ejpam-4835	45	19	establish	establish	VERB
ejpam-4835	45	20	a	a	DET
ejpam-4835	45	21	series	series	NOUN
ejpam-4835	45	22	of	of	ADP
ejpam-4835	45	23	uniform	uniform	ADJ
ejpam-4835	45	24	regular	regular	ADJ
ejpam-4835	45	25	triangulations	triangulation	NOUN
ejpam-4835	45	26	denoted	denote	VERB
ejpam-4835	45	27	as	as	ADP
ejpam-4835	45	28	{	{	PUNCT
ejpam-4835	45	29	tk	tk	PROPN
ejpam-4835	45	30	,	,	PUNCT
ejpam-4835	45	31	k	k	PROPN
ejpam-4835	45	32	∈	∈	PROPN
ejpam-4835	45	33	n0	n0	PROPN
ejpam-4835	45	34	}	}	PUNCT
ejpam-4835	45	35	.	.	PUNCT
ejpam-4835	46	1	for	for	ADP
ejpam-4835	46	2	all	all	DET
ejpam-4835	46	3	tk	tk	PROPN
ejpam-4835	46	4	,	,	PUNCT
ejpam-4835	46	5	we	we	PRON
ejpam-4835	46	6	have	have	VERB
ejpam-4835	46	7	the	the	DET
ejpam-4835	46	8	following	follow	VERB
ejpam-4835	46	9	estimates	estimate	NOUN
ejpam-4835	46	10	:	:	PUNCT
ejpam-4835	46	11	ωk	ωk	ADP
ejpam-4835	46	12	⊂	⊂	PROPN
ejpam-4835	46	13	ωk+1	ωk+1	NUM
ejpam-4835	46	14	⊂	⊂	PROPN
ejpam-4835	46	15	ω	ω	PROPN
ejpam-4835	46	16	,	,	PUNCT
ejpam-4835	46	17	dist	dist	NOUN
ejpam-4835	46	18	(	(	PUNCT
ejpam-4835	46	19	∂ωk	∂ωk	PROPN
ejpam-4835	46	20	,	,	PUNCT
ejpam-4835	46	21	∂ω	∂ω	ADJ
ejpam-4835	46	22	)	)	PUNCT
ejpam-4835	46	23	≤	≤	NUM
ejpam-4835	46	24	c0h	c0h	X
ejpam-4835	46	25	2	2	NUM
ejpam-4835	46	26	k	k	NOUN
ejpam-4835	46	27	,	,	PUNCT
ejpam-4835	46	28	hkhk+1	hkhk+1	VERB
ejpam-4835	46	29	≤	≤	PROPN
ejpam-4835	46	30	c1	c1	NOUN
ejpam-4835	46	31	.	.	PUNCT
ejpam-4835	47	1	we	we	PRON
ejpam-4835	47	2	introduce	introduce	VERB
ejpam-4835	47	3	vhk	vhk	ADJ
ejpam-4835	47	4	=	=	SYM
ejpam-4835	47	5	{	{	PUNCT
ejpam-4835	47	6	vhk	vhk	PROPN
ejpam-4835	47	7	∈	∈	PROPN
ejpam-4835	47	8	c(ω	c(ω	PROPN
ejpam-4835	47	9	)	)	PUNCT
ejpam-4835	47	10	∩h1	∩h1	PROPN
ejpam-4835	48	1	so	so	ADV
ejpam-4835	48	2	vhk	vhk	ADJ
ejpam-4835	48	3	/t	/t	SYM
ejpam-4835	48	4	∈	∈	PROPN
ejpam-4835	48	5	p1	p1	NOUN
ejpam-4835	48	6	}	}	PUNCT
ejpam-4835	48	7	,	,	PUNCT
ejpam-4835	48	8	we	we	PRON
ejpam-4835	48	9	simply	simply	ADV
ejpam-4835	48	10	write	write	VERB
ejpam-4835	48	11	:	:	PUNCT
ejpam-4835	48	12	vk	vk	PROPN
ejpam-4835	48	13	=	=	SYM
ejpam-4835	48	14	{	{	PUNCT
ejpam-4835	48	15	vk	vk	PROPN
ejpam-4835	48	16	∈	∈	PROPN
ejpam-4835	48	17	c(ω	c(ω	PROPN
ejpam-4835	48	18	)	)	PUNCT
ejpam-4835	49	1	∩h1	∩h1	NOUN
ejpam-4835	49	2	this	this	DET
ejpam-4835	49	3	vk	vk	NOUN
ejpam-4835	49	4	/	/	SYM
ejpam-4835	49	5	r	r	NOUN
ejpam-4835	49	6	∈	∈	PROPN
ejpam-4835	49	7	p1	p1	NOUN
ejpam-4835	49	8	}	}	PUNCT
ejpam-4835	49	9	.	.	PUNCT
ejpam-4835	50	1	h.	h.	PROPN
ejpam-4835	50	2	ahmad	ahmad	PROPN
ejpam-4835	50	3	et	et	PROPN
ejpam-4835	50	4	al	al	PROPN
ejpam-4835	50	5	.	.	PUNCT
ejpam-4835	50	6	/	/	SYM
ejpam-4835	50	7	eur	eur	PROPN
ejpam-4835	50	8	.	.	PUNCT
ejpam-4835	51	1	j.	j.	PROPN
ejpam-4835	51	2	pure	pure	PROPN
ejpam-4835	51	3	appl	appl	PROPN
ejpam-4835	51	4	.	.	PROPN
ejpam-4835	51	5	math	math	PROPN
ejpam-4835	51	6	,	,	PUNCT
ejpam-4835	51	7	16	16	NUM
ejpam-4835	51	8	(	(	PUNCT
ejpam-4835	51	9	3	3	NUM
ejpam-4835	51	10	)	)	PUNCT
ejpam-4835	51	11	(	(	PUNCT
ejpam-4835	51	12	2023	2023	NUM
ejpam-4835	51	13	)	)	PUNCT
ejpam-4835	51	14	,	,	PUNCT
ejpam-4835	51	15	1956	1956	NUM
ejpam-4835	51	16	-	-	SYM
ejpam-4835	51	17	1969	1969	NUM
ejpam-4835	51	18	1959	1959	NUM
ejpam-4835	51	19	the	the	DET
ejpam-4835	51	20	function	function	NOUN
ejpam-4835	51	21	of	of	ADP
ejpam-4835	51	22	the	the	DET
ejpam-4835	51	23	usual	usual	ADJ
ejpam-4835	51	24	basis	basis	NOUN
ejpam-4835	51	25	φi	φi	ADP
ejpam-4835	51	26	k	k	PROPN
ejpam-4835	51	27	,	,	PUNCT
ejpam-4835	51	28	i	i	PRON
ejpam-4835	51	29	∈	∈	PROPN
ejpam-4835	51	30	(	(	PUNCT
ejpam-4835	51	31	1	1	NUM
ejpam-4835	51	32	,	,	PUNCT
ejpam-4835	51	33	.	.	PUNCT
ejpam-4835	51	34	.	.	PUNCT
ejpam-4835	52	1	.	.	PUNCT
ejpam-4835	53	1	.,m	.,m	PUNCT
ejpam-4835	53	2	(	(	PUNCT
ejpam-4835	53	3	hk	hk	PROPN
ejpam-4835	53	4	)	)	PUNCT
ejpam-4835	53	5	)	)	PUNCT
ejpam-4835	53	6	is	be	AUX
ejpam-4835	53	7	defined	define	VERB
ejpam-4835	53	8	as	as	ADP
ejpam-4835	53	9	:	:	PUNCT
ejpam-4835	53	10	φi	φi	ADP
ejpam-4835	53	11	k	k	X
ejpam-4835	53	12	(	(	PUNCT
ejpam-4835	53	13	xjk	xjk	X
ejpam-4835	53	14	)	)	PUNCT
ejpam-4835	53	15	=	=	SYM
ejpam-4835	53	16	δij	δij	NOUN
ejpam-4835	53	17	,	,	PUNCT
ejpam-4835	53	18	x	x	PROPN
ejpam-4835	53	19	j	j	PROPN
ejpam-4835	53	20	k	k	PROPN
ejpam-4835	53	21	is	be	AUX
ejpam-4835	53	22	a	a	DET
ejpam-4835	53	23	node	node	NOUN
ejpam-4835	53	24	of	of	ADP
ejpam-4835	53	25	the	the	DET
ejpam-4835	53	26	triangulation	triangulation	NOUN
ejpam-4835	53	27	tk	tk	NOUN
ejpam-4835	53	28	.	.	PUNCT
ejpam-4835	54	1	we	we	PRON
ejpam-4835	54	2	define	define	VERB
ejpam-4835	54	3	the	the	DET
ejpam-4835	54	4	normal	normal	ADJ
ejpam-4835	54	5	restriction	restriction	NOUN
ejpam-4835	54	6	operator	operator	NOUN
ejpam-4835	54	7	as	as	SCONJ
ejpam-4835	54	8	follows	follow	VERB
ejpam-4835	54	9	:	:	PUNCT
ejpam-4835	54	10	rkv(x	rkv(x	PROPN
ejpam-4835	54	11	)	)	PUNCT
ejpam-4835	55	1	=	=	PUNCT
ejpam-4835	55	2	m(hk)∑	m(hk)∑	PROPN
ejpam-4835	55	3	i=1	i=1	PROPN
ejpam-4835	55	4	v	v	PROPN
ejpam-4835	55	5	(	(	PUNCT
ejpam-4835	55	6	m	m	VERB
ejpam-4835	55	7	i	i	NOUN
ejpam-4835	55	8	k	k	NOUN
ejpam-4835	55	9	)	)	PUNCT
ejpam-4835	55	10	φi	φi	ADP
ejpam-4835	55	11	k(x	k(x	PROPN
ejpam-4835	55	12	)	)	PUNCT
ejpam-4835	55	13	.	.	PUNCT
ejpam-4835	56	1	(	(	PUNCT
ejpam-4835	56	2	2.4	2.4	NUM
ejpam-4835	56	3	)	)	PUNCT
ejpam-4835	56	4	if	if	SCONJ
ejpam-4835	56	5	we	we	PRON
ejpam-4835	56	6	write	write	VERB
ejpam-4835	56	7	uk	uk	PROPN
ejpam-4835	56	8	=	=	PROPN
ejpam-4835	56	9	rmk	rmk	PROPN
ejpam-4835	56	10	,	,	PUNCT
ejpam-4835	56	11	rk	rk	NOUN
ejpam-4835	56	12	is	be	AUX
ejpam-4835	56	13	a	a	DET
ejpam-4835	56	14	bijection	bijection	NOUN
ejpam-4835	56	15	from	from	ADP
ejpam-4835	56	16	uk	uk	PROPN
ejpam-4835	56	17	to	to	PART
ejpam-4835	56	18	vk	vk	VERB
ejpam-4835	56	19	.	.	PUNCT
ejpam-4835	57	1	in	in	ADP
ejpam-4835	57	2	uk	uk	PROPN
ejpam-4835	57	3	,	,	PUNCT
ejpam-4835	57	4	we	we	PRON
ejpam-4835	57	5	define	define	VERB
ejpam-4835	57	6	the	the	DET
ejpam-4835	57	7	scalar	scalar	ADJ
ejpam-4835	57	8	product	product	NOUN
ejpam-4835	57	9	:	:	PUNCT
ejpam-4835	57	10	<	<	X
ejpam-4835	57	11	u	u	NOUN
ejpam-4835	57	12	,	,	PUNCT
ejpam-4835	57	13	v	v	X
ejpam-4835	57	14	>	>	X
ejpam-4835	57	15	=	=	PUNCT
ejpam-4835	58	1	h2k	h2k	ADJ
ejpam-4835	58	2	m(hk)∑	m(hk)∑	PROPN
ejpam-4835	58	3	i=1	i=1	PROPN
ejpam-4835	58	4	uivi	uivi	PROPN
ejpam-4835	58	5	,	,	PUNCT
ejpam-4835	58	6	∥u∥k	∥u∥k	AUX
ejpam-4835	58	7	=	=	SYM
ejpam-4835	58	8	<	<	X
ejpam-4835	58	9	u	u	NOUN
ejpam-4835	58	10	,	,	PUNCT
ejpam-4835	58	11	v	v	X
ejpam-4835	58	12	>	>	X
ejpam-4835	58	13	1/2	1/2	NUM
ejpam-4835	58	14	k	k	NOUN
ejpam-4835	58	15	.	.	PUNCT
ejpam-4835	59	1	the	the	DET
ejpam-4835	59	2	maximum	maximum	ADJ
ejpam-4835	59	3	norm	norm	NOUN
ejpam-4835	59	4	∥	∥	X
ejpam-4835	59	5	·	·	SYM
ejpam-4835	59	6	∥∞	∥∞	PROPN
ejpam-4835	59	7	(	(	PUNCT
ejpam-4835	59	8	in	in	ADP
ejpam-4835	59	9	uk	uk	PROPN
ejpam-4835	59	10	)	)	PUNCT
ejpam-4835	59	11	and	and	CCONJ
ejpam-4835	59	12	∥	∥	NUM
ejpam-4835	59	13	·	·	PUNCT
ejpam-4835	59	14	∥l∞	∥l∞	PUNCT
ejpam-4835	59	15	(	(	PUNCT
ejpam-4835	59	16	in	in	ADP
ejpam-4835	59	17	vk	vk	NOUN
ejpam-4835	59	18	)	)	PUNCT
ejpam-4835	59	19	are	be	AUX
ejpam-4835	59	20	equivalent	equivalent	ADJ
ejpam-4835	59	21	standards	standard	NOUN
ejpam-4835	59	22	,	,	PUNCT
ejpam-4835	59	23	we	we	PRON
ejpam-4835	59	24	denote	denote	VERB
ejpam-4835	59	25	them	they	PRON
ejpam-4835	59	26	∥	∥	X
ejpam-4835	59	27	·	·	PUNCT
ejpam-4835	59	28	∥∞.	∥∞.	INTJ
ejpam-4835	59	29	we	we	PRON
ejpam-4835	59	30	have	have	VERB
ejpam-4835	59	31	the	the	DET
ejpam-4835	59	32	following	follow	VERB
ejpam-4835	59	33	lemma	lemma	PROPN
ejpam-4835	59	34	(	(	PUNCT
ejpam-4835	59	35	see	see	VERB
ejpam-4835	59	36	[	[	X
ejpam-4835	59	37	11	11	NUM
ejpam-4835	59	38	]	]	NUM
ejpam-4835	59	39	)	)	PUNCT
ejpam-4835	59	40	.	.	PUNCT
ejpam-4835	60	1	lemma	lemma	PROPN
ejpam-4835	60	2	1	1	NUM
ejpam-4835	60	3	:	:	PUNCT
ejpam-4835	60	4	there	there	PRON
ejpam-4835	60	5	exists	exist	VERB
ejpam-4835	60	6	c1	c1	PROPN
ejpam-4835	60	7	,	,	PUNCT
ejpam-4835	60	8	c2	c2	PROPN
ejpam-4835	60	9	independent	independent	ADJ
ejpam-4835	60	10	of	of	ADP
ejpam-4835	60	11	k	k	PROPN
ejpam-4835	60	12	so	so	ADV
ejpam-4835	60	13	:	:	PUNCT
ejpam-4835	60	14	∥rk(u)∥l∞	∥rk(u)∥l∞	NOUN
ejpam-4835	60	15	=	=	SYM
ejpam-4835	60	16	∥u∥l∞	∥u∥l∞	NOUN
ejpam-4835	60	17	,	,	PUNCT
ejpam-4835	60	18	∀u	∀u	NOUN
ejpam-4835	60	19	∈	∈	PROPN
ejpam-4835	60	20	uk	uk	PROPN
ejpam-4835	60	21	.	.	PROPN
ejpam-4835	60	22	c1∥v∥l∞	c1∥v∥l∞	PROPN
ejpam-4835	61	1	≤	≤	X
ejpam-4835	62	1	∥r∗k(v)∥l∞	∥r∗k(v)∥l∞	X
ejpam-4835	62	2	≤	≤	NUM
ejpam-4835	62	3	c2∥	c2∥	PROPN
ejpam-4835	62	4	v∥l∞	v∥l∞	PROPN
ejpam-4835	62	5	,	,	PUNCT
ejpam-4835	62	6	∀v	∀v	PROPN
ejpam-4835	62	7	∈	∈	PROPN
ejpam-4835	62	8	vk	vk	X
ejpam-4835	62	9	.	.	PUNCT
ejpam-4835	63	1	(	(	PUNCT
ejpam-4835	63	2	2.5	2.5	NUM
ejpam-4835	63	3	)	)	PUNCT
ejpam-4835	63	4	2.4	2.4	NUM
ejpam-4835	63	5	.	.	PUNCT
ejpam-4835	64	1	discrete	discrete	ADJ
ejpam-4835	64	2	problem	problem	NOUN
ejpam-4835	64	3	in	in	ADP
ejpam-4835	64	4	a	a	DET
ejpam-4835	64	5	natural	natural	ADJ
ejpam-4835	64	6	progression	progression	NOUN
ejpam-4835	64	7	,	,	PUNCT
ejpam-4835	64	8	we	we	PRON
ejpam-4835	64	9	introduce	introduce	VERB
ejpam-4835	64	10	the	the	DET
ejpam-4835	64	11	discretization	discretization	NOUN
ejpam-4835	64	12	matrices	matrice	VERB
ejpam-4835	64	13	ak	ak	PROPN
ejpam-4835	64	14	and	and	CCONJ
ejpam-4835	64	15	the	the	DET
ejpam-4835	64	16	generic	generic	ADJ
ejpam-4835	64	17	coefficient	coefficient	NOUN
ejpam-4835	64	18	matrices	matrice	VERB
ejpam-4835	64	19	a	a	DET
ejpam-4835	64	20	(	(	PUNCT
ejpam-4835	64	21	φk1	φk1	NOUN
ejpam-4835	64	22	,	,	PUNCT
ejpam-4835	64	23	φs	φs	ADP
ejpam-4835	64	24	k	k	PROPN
ejpam-4835	64	25	)	)	PUNCT
ejpam-4835	64	26	,	,	PUNCT
ejpam-4835	64	27	where	where	SCONJ
ejpam-4835	64	28	φs	φs	ADV
ejpam-4835	64	29	,	,	PUNCT
ejpam-4835	64	30	s	s	PART
ejpam-4835	64	31	=	=	SYM
ejpam-4835	64	32	1	1	NUM
ejpam-4835	64	33	,	,	PUNCT
ejpam-4835	64	34	2	2	NUM
ejpam-4835	64	35	,	,	PUNCT
ejpam-4835	64	36	.	.	PUNCT
ejpam-4835	64	37	.	.	PUNCT
ejpam-4835	65	1	.m	.m	PROPN
ejpam-4835	66	1	(	(	PUNCT
ejpam-4835	66	2	hk	hk	PROPN
ejpam-4835	66	3	)	)	PUNCT
ejpam-4835	66	4	represent	represent	VERB
ejpam-4835	66	5	the	the	DET
ejpam-4835	66	6	customary	customary	ADJ
ejpam-4835	66	7	basic	basic	ADJ
ejpam-4835	66	8	functions	function	NOUN
ejpam-4835	66	9	.	.	PUNCT
ejpam-4835	67	1	with	with	ADP
ejpam-4835	67	2	these	these	DET
ejpam-4835	67	3	definitions	definition	NOUN
ejpam-4835	67	4	in	in	ADP
ejpam-4835	67	5	place	place	NOUN
ejpam-4835	67	6	,	,	PUNCT
ejpam-4835	67	7	we	we	PRON
ejpam-4835	67	8	can	can	AUX
ejpam-4835	67	9	now	now	ADV
ejpam-4835	67	10	define	define	VERB
ejpam-4835	67	11	the	the	DET
ejpam-4835	67	12	discrete	discrete	ADJ
ejpam-4835	67	13	problem	problem	NOUN
ejpam-4835	67	14	as	as	SCONJ
ejpam-4835	67	15	follows	follow	VERB
ejpam-4835	67	16	:	:	PUNCT
ejpam-4835	67	17	find	find	VERB
ejpam-4835	67	18	uk	uk	PROPN
ejpam-4835	67	19	∈	∈	PROPN
ejpam-4835	67	20	vk	vk	NOUN
ejpam-4835	67	21	,	,	PUNCT
ejpam-4835	67	22	which	which	PRON
ejpam-4835	67	23	serves	serve	VERB
ejpam-4835	67	24	as	as	ADP
ejpam-4835	67	25	the	the	DET
ejpam-4835	67	26	solution	solution	NOUN
ejpam-4835	67	27	of	of	ADP
ejpam-4835	67	28	:	:	PUNCT
ejpam-4835	67	29	{	{	PUNCT
ejpam-4835	67	30	<	<	X
ejpam-4835	67	31	akuk	akuk	X
ejpam-4835	67	32	,	,	PUNCT
ejpam-4835	67	33	vk	vk	ADP
ejpam-4835	67	34	−	−	PROPN
ejpam-4835	67	35	uk	uk	PROPN
ejpam-4835	67	36	>	>	PUNCT
ejpam-4835	67	37	≥	≥	X
ejpam-4835	67	38	<	<	X
ejpam-4835	67	39	fk(uk	fk(uk	PROPN
ejpam-4835	67	40	)	)	PUNCT
ejpam-4835	67	41	,	,	PUNCT
ejpam-4835	67	42	vk	vk	ADP
ejpam-4835	67	43	−	−	PROPN
ejpam-4835	67	44	uk	uk	PROPN
ejpam-4835	67	45	>	>	PUNCT
ejpam-4835	67	46	,	,	PUNCT
ejpam-4835	67	47	∀vk	∀vk	PROPN
ejpam-4835	67	48	∈	∈	PROPN
ejpam-4835	67	49	vk	vk	NOUN
ejpam-4835	67	50	.	.	PUNCT
ejpam-4835	68	1	uk	uk	PROPN
ejpam-4835	68	2	≤	≤	PROPN
ejpam-4835	68	3	rkψ	rkψ	PROPN
ejpam-4835	68	4	;	;	PUNCT
ejpam-4835	68	5	vk	vk	PROPN
ejpam-4835	68	6	≤	≤	NUM
ejpam-4835	68	7	rkψ	rkψ	PROPN
ejpam-4835	68	8	.	.	PUNCT
ejpam-4835	69	1	(	(	PUNCT
ejpam-4835	69	2	2.6	2.6	NUM
ejpam-4835	69	3	)	)	PUNCT
ejpam-4835	69	4	we	we	PRON
ejpam-4835	69	5	assume	assume	VERB
ejpam-4835	69	6	that	that	SCONJ
ejpam-4835	69	7	the	the	DET
ejpam-4835	69	8	matrices	matrix	NOUN
ejpam-4835	69	9	ak	ak	PROPN
ejpam-4835	69	10	are	be	AUX
ejpam-4835	69	11	m	m	NOUN
ejpam-4835	69	12	-	-	NOUN
ejpam-4835	69	13	matrices	matrix	NOUN
ejpam-4835	69	14	(	(	PUNCT
ejpam-4835	69	15	see	see	VERB
ejpam-4835	69	16	[	[	X
ejpam-4835	69	17	12	12	NUM
ejpam-4835	69	18	]	]	NUM
ejpam-4835	69	19	)	)	PUNCT
ejpam-4835	69	20	.	.	PUNCT
ejpam-4835	70	1	2.5	2.5	NUM
ejpam-4835	70	2	.	.	PUNCT
ejpam-4835	70	3	formulation	formulation	NOUN
ejpam-4835	70	4	in	in	ADP
ejpam-4835	70	5	hjb	hjb	NOUN
ejpam-4835	70	6	:	:	PUNCT
ejpam-4835	70	7	the	the	DET
ejpam-4835	70	8	equivalence	equivalence	NOUN
ejpam-4835	70	9	between	between	ADP
ejpam-4835	70	10	the	the	DET
ejpam-4835	70	11	finite	finite	ADJ
ejpam-4835	70	12	-	-	ADJ
ejpam-4835	70	13	dimensional	dimensional	ADJ
ejpam-4835	70	14	variational	variational	ADJ
ejpam-4835	70	15	inequality	inequality	NOUN
ejpam-4835	70	16	(	(	PUNCT
ejpam-4835	70	17	2.3	2.3	NUM
ejpam-4835	70	18	)	)	PUNCT
ejpam-4835	70	19	and	and	CCONJ
ejpam-4835	70	20	a	a	DET
ejpam-4835	70	21	formulation	formulation	NOUN
ejpam-4835	70	22	in	in	ADP
ejpam-4835	70	23	hamilton	hamilton	PROPN
ejpam-4835	70	24	-	-	PUNCT
ejpam-4835	70	25	jacobi	jacobi	PROPN
ejpam-4835	70	26	-	-	PUNCT
ejpam-4835	70	27	bellman	bellman	NOUN
ejpam-4835	70	28	(	(	PUNCT
ejpam-4835	70	29	hjb	hjb	NOUN
ejpam-4835	70	30	)	)	PUNCT
ejpam-4835	70	31	can	can	AUX
ejpam-4835	70	32	be	be	AUX
ejpam-4835	70	33	readily	readily	ADV
ejpam-4835	70	34	observed	observe	VERB
ejpam-4835	70	35	(	(	PUNCT
ejpam-4835	70	36	see	see	VERB
ejpam-4835	70	37	[	[	X
ejpam-4835	70	38	19	19	NUM
ejpam-4835	70	39	]	]	NUM
ejpam-4835	70	40	)	)	PUNCT
ejpam-4835	70	41	.	.	PUNCT
ejpam-4835	71	1	we	we	PRON
ejpam-4835	71	2	describe	describe	VERB
ejpam-4835	71	3	the	the	DET
ejpam-4835	71	4	numerical	numerical	ADJ
ejpam-4835	71	5	algorithm	algorithm	NOUN
ejpam-4835	71	6	chosen	choose	VERB
ejpam-4835	71	7	to	to	PART
ejpam-4835	71	8	solve	solve	VERB
ejpam-4835	71	9	the	the	DET
ejpam-4835	71	10	stationary	stationary	ADJ
ejpam-4835	71	11	hjb	hjb	NOUN
ejpam-4835	71	12	equations	equation	NOUN
ejpam-4835	71	13	.	.	PUNCT
ejpam-4835	72	1	in	in	ADP
ejpam-4835	72	2	the	the	DET
ejpam-4835	72	3	classic	classic	ADJ
ejpam-4835	72	4	framework	framework	NOUN
ejpam-4835	72	5	,	,	PUNCT
ejpam-4835	72	6	we	we	PRON
ejpam-4835	72	7	recall	recall	VERB
ejpam-4835	72	8	some	some	DET
ejpam-4835	72	9	convergence	convergence	NOUN
ejpam-4835	72	10	results	result	NOUN
ejpam-4835	72	11	that	that	PRON
ejpam-4835	72	12	will	will	AUX
ejpam-4835	72	13	be	be	AUX
ejpam-4835	72	14	instrumental	instrumental	ADJ
ejpam-4835	72	15	in	in	ADP
ejpam-4835	72	16	establishing	establish	VERB
ejpam-4835	72	17	the	the	DET
ejpam-4835	72	18	convergence	convergence	NOUN
ejpam-4835	72	19	of	of	ADP
ejpam-4835	72	20	the	the	DET
ejpam-4835	72	21	mghjb	mghjb	NOUN
ejpam-4835	72	22	algorithm	algorithm	PROPN
ejpam-4835	72	23	described	describe	VERB
ejpam-4835	72	24	in	in	ADP
ejpam-4835	72	25	the	the	DET
ejpam-4835	72	26	subsequent	subsequent	ADJ
ejpam-4835	72	27	section	section	NOUN
ejpam-4835	72	28	.	.	PUNCT
ejpam-4835	73	1	iterative	iterative	NOUN
ejpam-4835	73	2	diagram	diagram	PROPN
ejpam-4835	73	3	:	:	PUNCT
ejpam-4835	73	4	step	step	NOUN
ejpam-4835	73	5	1	1	NUM
ejpam-4835	73	6	:	:	PUNCT
ejpam-4835	73	7	choose	choose	VERB
ejpam-4835	73	8	an	an	DET
ejpam-4835	73	9	initial	initial	ADJ
ejpam-4835	73	10	vector	vector	NOUN
ejpam-4835	73	11	u0k	u0k	PROPN
ejpam-4835	73	12	∈	∈	PROPN
ejpam-4835	73	13	rnk	rnk	X
ejpam-4835	73	14	.	.	PUNCT
ejpam-4835	74	1	step	step	NOUN
ejpam-4835	74	2	2	2	NUM
ejpam-4835	74	3	:	:	PUNCT
ejpam-4835	74	4	let	let	VERB
ejpam-4835	74	5	u	u	PRON
ejpam-4835	74	6	(	(	PUNCT
ejpam-4835	74	7	ν	ν	NOUN
ejpam-4835	74	8	)	)	PUNCT
ejpam-4835	74	9	k	k	PROPN
ejpam-4835	74	10	∈	∈	PROPN
ejpam-4835	74	11	rnk	rnk	X
ejpam-4835	74	12	,	,	PUNCT
ejpam-4835	74	13	ν	ν	X
ejpam-4835	74	14	≥	≥	NOUN
ejpam-4835	74	15	0	0	NUM
ejpam-4835	74	16	,	,	PUNCT
ejpam-4835	74	17	and	and	CCONJ
ejpam-4835	74	18	calculate	calculate	VERB
ejpam-4835	74	19	u	u	PROPN
ejpam-4835	74	20	(	(	PUNCT
ejpam-4835	74	21	ν+1	ν+1	PROPN
ejpam-4835	74	22	)	)	PUNCT
ejpam-4835	74	23	k	k	PROPN
ejpam-4835	74	24	∈	∈	PROPN
ejpam-4835	74	25	rnk	rnk	X
ejpam-4835	74	26	as	as	ADP
ejpam-4835	74	27	the	the	DET
ejpam-4835	74	28	solution	solution	NOUN
ejpam-4835	74	29	of	of	ADP
ejpam-4835	74	30	the	the	DET
ejpam-4835	74	31	equation	equation	NOUN
ejpam-4835	74	32	:	:	PUNCT
ejpam-4835	74	33	aν	aν	PROPN
ejpam-4835	74	34	ku	ku	PROPN
ejpam-4835	74	35	ν+1	ν+1	PROPN
ejpam-4835	75	1	k	k	PROPN
ejpam-4835	76	1	−	−	PROPN
ejpam-4835	76	2	zν	zν	INTJ
ejpam-4835	76	3	k	k	NOUN
ejpam-4835	76	4	=	=	SYM
ejpam-4835	76	5	0	0	PROPN
ejpam-4835	76	6	,	,	PUNCT
ejpam-4835	76	7	(	(	PUNCT
ejpam-4835	76	8	2.7	2.7	NUM
ejpam-4835	76	9	)	)	PUNCT
ejpam-4835	76	10	h.	h.	PROPN
ejpam-4835	76	11	ahmad	ahmad	PROPN
ejpam-4835	76	12	et	et	PROPN
ejpam-4835	76	13	al	al	PROPN
ejpam-4835	76	14	.	.	PUNCT
ejpam-4835	76	15	/	/	SYM
ejpam-4835	76	16	eur	eur	PROPN
ejpam-4835	76	17	.	.	PUNCT
ejpam-4835	77	1	j.	j.	PROPN
ejpam-4835	77	2	pure	pure	PROPN
ejpam-4835	77	3	appl	appl	PROPN
ejpam-4835	77	4	.	.	PROPN
ejpam-4835	77	5	math	math	PROPN
ejpam-4835	77	6	,	,	PUNCT
ejpam-4835	77	7	16	16	NUM
ejpam-4835	77	8	(	(	PUNCT
ejpam-4835	77	9	3	3	NUM
ejpam-4835	77	10	)	)	PUNCT
ejpam-4835	77	11	(	(	PUNCT
ejpam-4835	77	12	2023	2023	NUM
ejpam-4835	77	13	)	)	PUNCT
ejpam-4835	77	14	,	,	PUNCT
ejpam-4835	77	15	1956	1956	NUM
ejpam-4835	77	16	-	-	SYM
ejpam-4835	77	17	1969	1969	NUM
ejpam-4835	77	18	1960	1960	NUM
ejpam-4835	77	19	such	such	ADJ
ejpam-4835	77	20	that	that	PRON
ejpam-4835	77	21	:	:	PUNCT
ejpam-4835	77	22	zν	zν	X
ejpam-4835	77	23	k	k	NOUN
ejpam-4835	77	24	=	=	PUNCT
ejpam-4835	78	1	f	f	PROPN
ejpam-4835	78	2	ν	ν	X
ejpam-4835	78	3	k	k	X
ejpam-4835	78	4	(	(	PUNCT
ejpam-4835	78	5	u	u	NOUN
ejpam-4835	78	6	ν	ν	X
ejpam-4835	78	7	k	k	PROPN
ejpam-4835	78	8	)	)	PUNCT
ejpam-4835	78	9	,	,	PUNCT
ejpam-4835	78	10	where	where	SCONJ
ejpam-4835	78	11	:	:	PUNCT
ejpam-4835	78	12	aν	aν	NOUN
ejpam-4835	78	13	k	k	NOUN
ejpam-4835	78	14	,	,	PUNCT
ejpam-4835	78	15	i	i	PRON
ejpam-4835	78	16	=	=	PRON
ejpam-4835	78	17	{	{	PUNCT
ejpam-4835	78	18	ak	ak	PROPN
ejpam-4835	78	19	,	,	PUNCT
ejpam-4835	78	20	i	i	PRON
ejpam-4835	78	21	(	(	PUNCT
ejpam-4835	78	22	uk	uk	PROPN
ejpam-4835	78	23	)	)	PUNCT
ejpam-4835	78	24	if	if	SCONJ
ejpam-4835	78	25	ak	ak	PROPN
ejpam-4835	78	26	,	,	PUNCT
ejpam-4835	78	27	iu	iu	ADP
ejpam-4835	78	28	ν	ν	X
ejpam-4835	78	29	k	k	PROPN
ejpam-4835	78	30	,	,	PUNCT
ejpam-4835	78	31	i	i	PRON
ejpam-4835	78	32	−	−	PROPN
ejpam-4835	78	33	zk	zk	PROPN
ejpam-4835	78	34	,	,	PUNCT
ejpam-4835	78	35	i	i	PRON
ejpam-4835	78	36	>	>	X
ejpam-4835	78	37	uνk	uνk	PROPN
ejpam-4835	78	38	,	,	PUNCT
ejpam-4835	78	39	i	i	PRON
ejpam-4835	78	40	−	−	VERB
ejpam-4835	78	41	ψk	ψk	VERB
ejpam-4835	78	42	,	,	PUNCT
ejpam-4835	78	43	i.	i.	PROPN
ejpam-4835	78	44	uk	uk	PROPN
ejpam-4835	78	45	,	,	PUNCT
ejpam-4835	78	46	i	i	PRON
ejpam-4835	78	47	if	if	SCONJ
ejpam-4835	78	48	1	1	NUM
ejpam-4835	78	49	≤	≤	PUNCT
ejpam-4835	78	50	i	i	NOUN
ejpam-4835	78	51	≤	≤	NOUN
ejpam-4835	78	52	n	n	ADV
ejpam-4835	78	53	.	.	PUNCT
ejpam-4835	79	1	(	(	PUNCT
ejpam-4835	79	2	2.8a	2.8a	NOUN
ejpam-4835	79	3	)	)	PUNCT
ejpam-4835	79	4	zν	zν	NOUN
ejpam-4835	80	1	k	k	NOUN
ejpam-4835	80	2	,	,	PUNCT
ejpam-4835	80	3	i	i	PRON
ejpam-4835	80	4	=	=	PRON
ejpam-4835	80	5	{	{	PUNCT
ejpam-4835	80	6	zk	zk	PROPN
ejpam-4835	80	7	,	,	PUNCT
ejpam-4835	80	8	i	i	PRON
ejpam-4835	80	9	if	if	SCONJ
ejpam-4835	80	10	ak	ak	PROPN
ejpam-4835	80	11	,	,	PUNCT
ejpam-4835	80	12	iu	iu	ADP
ejpam-4835	80	13	ν	ν	X
ejpam-4835	80	14	k	k	PROPN
ejpam-4835	80	15	,	,	PUNCT
ejpam-4835	80	16	i	i	PRON
ejpam-4835	80	17	−	−	PROPN
ejpam-4835	80	18	zk	zk	PROPN
ejpam-4835	80	19	,	,	PUNCT
ejpam-4835	80	20	i	i	PRON
ejpam-4835	80	21	>	>	X
ejpam-4835	80	22	uνk	uνk	PROPN
ejpam-4835	80	23	,	,	PUNCT
ejpam-4835	80	24	i	i	PRON
ejpam-4835	80	25	−	−	VERB
ejpam-4835	80	26	ψk	ψk	VERB
ejpam-4835	80	27	,	,	PUNCT
ejpam-4835	80	28	i	i	PRON
ejpam-4835	80	29	uk	uk	PROPN
ejpam-4835	80	30	,	,	PUNCT
ejpam-4835	80	31	i	i	PRON
ejpam-4835	80	32	if	if	SCONJ
ejpam-4835	80	33	1	1	NUM
ejpam-4835	80	34	≤	≤	PUNCT
ejpam-4835	80	35	i	i	NOUN
ejpam-4835	80	36	≤	≤	ADJ
ejpam-4835	81	1	n	n	CCONJ
ejpam-4835	81	2	(	(	PUNCT
ejpam-4835	81	3	2.8b	2.8b	NOUN
ejpam-4835	81	4	)	)	PUNCT
ejpam-4835	81	5	let	let	VERB
ejpam-4835	81	6	u∗k	u∗k	ADJ
ejpam-4835	81	7	be	be	AUX
ejpam-4835	81	8	the	the	DET
ejpam-4835	81	9	unique	unique	ADJ
ejpam-4835	81	10	solution	solution	NOUN
ejpam-4835	81	11	of	of	ADP
ejpam-4835	81	12	the	the	DET
ejpam-4835	81	13	discrete	discrete	ADJ
ejpam-4835	81	14	hjb	hjb	NOUN
ejpam-4835	81	15	equation	equation	NOUN
ejpam-4835	81	16	max	max	PROPN
ejpam-4835	81	17	1≤i≤n	1≤i≤n	NUM
ejpam-4835	81	18	(	(	PUNCT
ejpam-4835	81	19	ak	ak	PROPN
ejpam-4835	81	20	,	,	PUNCT
ejpam-4835	81	21	iu	iu	ADP
ejpam-4835	81	22	∗	∗	NOUN
ejpam-4835	81	23	k	k	PROPN
ejpam-4835	82	1	−	−	PROPN
ejpam-4835	82	2	zk	zk	PROPN
ejpam-4835	82	3	,	,	PUNCT
ejpam-4835	82	4	i	i	PRON
ejpam-4835	82	5	,	,	PUNCT
ejpam-4835	82	6	u	u	PROPN
ejpam-4835	82	7	∗	∗	NOUN
ejpam-4835	82	8	k	k	PROPN
ejpam-4835	82	9	,	,	PUNCT
ejpam-4835	82	10	i	i	PRON
ejpam-4835	82	11	−	−	VERB
ejpam-4835	82	12	ψk	ψk	VERB
ejpam-4835	82	13	,	,	PUNCT
ejpam-4835	82	14	i	i	PRON
ejpam-4835	82	15	)	)	PUNCT
ejpam-4835	83	1	=	=	PUNCT
ejpam-4835	83	2	0	0	X
ejpam-4835	83	3	.	.	PUNCT
ejpam-4835	84	1	(	(	PUNCT
ejpam-4835	84	2	2.9	2.9	NUM
ejpam-4835	84	3	)	)	PUNCT
ejpam-4835	84	4	we	we	PRON
ejpam-4835	84	5	shall	shall	AUX
ejpam-4835	84	6	state	state	VERB
ejpam-4835	84	7	the	the	DET
ejpam-4835	84	8	following	following	ADJ
ejpam-4835	84	9	theorem	theorem	NOUN
ejpam-4835	84	10	and	and	CCONJ
ejpam-4835	84	11	present	present	VERB
ejpam-4835	84	12	our	our	PRON
ejpam-4835	84	13	problem	problem	NOUN
ejpam-4835	84	14	formulated	formulate	VERB
ejpam-4835	84	15	based	base	VERB
ejpam-4835	84	16	on	on	ADP
ejpam-4835	84	17	the	the	DET
ejpam-4835	84	18	hamilton	hamilton	PROPN
ejpam-4835	84	19	-	-	PUNCT
ejpam-4835	84	20	jacobi	jacobi	PROPN
ejpam-4835	84	21	-	-	PUNCT
ejpam-4835	84	22	bellman	bellman	NOUN
ejpam-4835	84	23	(	(	PUNCT
ejpam-4835	84	24	hjb	hjb	NOUN
ejpam-4835	84	25	)	)	PUNCT
ejpam-4835	84	26	equation	equation	NOUN
ejpam-4835	84	27	,	,	PUNCT
ejpam-4835	84	28	which	which	PRON
ejpam-4835	84	29	is	be	AUX
ejpam-4835	84	30	an	an	DET
ejpam-4835	84	31	adaptation	adaptation	NOUN
ejpam-4835	84	32	of	of	ADP
ejpam-4835	84	33	hoppe	hoppe	PROPN
ejpam-4835	84	34	’s	’s	PART
ejpam-4835	84	35	work	work	NOUN
ejpam-4835	84	36	[	[	X
ejpam-4835	84	37	19	19	NUM
ejpam-4835	84	38	]	]	PUNCT
ejpam-4835	84	39	.	.	PUNCT
ejpam-4835	85	1	theorem	theorem	NOUN
ejpam-4835	85	2	2	2	NUM
ejpam-4835	85	3	:	:	PUNCT
ejpam-4835	85	4	let	let	VERB
ejpam-4835	85	5	uνk	uνk	ADV
ejpam-4835	85	6	be	be	AUX
ejpam-4835	85	7	the	the	DET
ejpam-4835	85	8	iteration	iteration	NOUN
ejpam-4835	85	9	defined	define	VERB
ejpam-4835	85	10	,	,	PUNCT
ejpam-4835	85	11	and	and	CCONJ
ejpam-4835	85	12	it	it	PRON
ejpam-4835	85	13	satisfies	satisfy	VERB
ejpam-4835	85	14	the	the	DET
ejpam-4835	85	15	hjb	hjb	NOUN
ejpam-4835	85	16	equation	equation	NOUN
ejpam-4835	85	17	.	.	PUNCT
ejpam-4835	86	1	additionally	additionally	ADV
ejpam-4835	86	2	,	,	PUNCT
ejpam-4835	86	3	we	we	PRON
ejpam-4835	86	4	assume	assume	VERB
ejpam-4835	86	5	that	that	SCONJ
ejpam-4835	86	6	ak	ak	PROPN
ejpam-4835	86	7	is	be	AUX
ejpam-4835	86	8	continuously	continuously	ADV
ejpam-4835	86	9	differentiable	differentiable	ADJ
ejpam-4835	86	10	.	.	PUNCT
ejpam-4835	87	1	under	under	ADP
ejpam-4835	87	2	these	these	DET
ejpam-4835	87	3	conditions	condition	NOUN
ejpam-4835	87	4	,	,	PUNCT
ejpam-4835	87	5	the	the	DET
ejpam-4835	87	6	sequence	sequence	NOUN
ejpam-4835	87	7	(	(	PUNCT
ejpam-4835	87	8	uνk)ν≥0	uνk)ν≥0	VERB
ejpam-4835	87	9	converges	converge	NOUN
ejpam-4835	87	10	and	and	CCONJ
ejpam-4835	87	11	approaches	approach	NOUN
ejpam-4835	87	12	u∗k	u∗k	ADJ
ejpam-4835	87	13	.	.	PUNCT
ejpam-4835	88	1	before	before	ADP
ejpam-4835	88	2	proceeding	proceed	VERB
ejpam-4835	88	3	to	to	PART
ejpam-4835	88	4	present	present	VERB
ejpam-4835	88	5	the	the	DET
ejpam-4835	88	6	results	result	NOUN
ejpam-4835	88	7	,	,	PUNCT
ejpam-4835	88	8	it	it	PRON
ejpam-4835	88	9	is	be	AUX
ejpam-4835	88	10	pertinent	pertinent	ADJ
ejpam-4835	88	11	to	to	PART
ejpam-4835	88	12	recall	recall	VERB
ejpam-4835	88	13	the	the	DET
ejpam-4835	88	14	following	follow	VERB
ejpam-4835	88	15	theorem	theorem	PROPN
ejpam-4835	88	16	.	.	PUNCT
ejpam-4835	88	17	theorem	theorem	NOUN
ejpam-4835	88	18	3	3	NUM
ejpam-4835	88	19	:	:	PUNCT
ejpam-4835	88	20	(	(	PUNCT
ejpam-4835	88	21	[	[	X
ejpam-4835	88	22	10	10	NUM
ejpam-4835	88	23	]	]	PUNCT
ejpam-4835	88	24	,	,	PUNCT
ejpam-4835	88	25	[	[	X
ejpam-4835	88	26	14	14	NUM
ejpam-4835	88	27	]	]	PUNCT
ejpam-4835	88	28	)	)	PUNCT
ejpam-4835	88	29	under	under	ADP
ejpam-4835	88	30	the	the	DET
ejpam-4835	88	31	previous	previous	ADJ
ejpam-4835	88	32	assumptions	assumption	NOUN
ejpam-4835	88	33	and	and	CCONJ
ejpam-4835	88	34	notation	notation	NOUN
ejpam-4835	88	35	,	,	PUNCT
ejpam-4835	88	36	we	we	PRON
ejpam-4835	88	37	have	have	VERB
ejpam-4835	88	38	:	:	PUNCT
ejpam-4835	88	39	∥u−	∥u−	VERB
ejpam-4835	88	40	u∗k∥∞	u∗k∥∞	PROPN
ejpam-4835	88	41	≤	≤	ADJ
ejpam-4835	88	42	ch2k	ch2k	PROPN
ejpam-4835	88	43	|loghk|	|loghk|	NUM
ejpam-4835	88	44	2	2	NUM
ejpam-4835	88	45	∥f(u)∥∞.	∥f(u)∥∞.	NOUN
ejpam-4835	88	46	(	(	PUNCT
ejpam-4835	88	47	2.10	2.10	NUM
ejpam-4835	88	48	)	)	PUNCT
ejpam-4835	88	49	2.6	2.6	NUM
ejpam-4835	88	50	.	.	PUNCT
ejpam-4835	89	1	description	description	NOUN
ejpam-4835	89	2	of	of	ADP
ejpam-4835	89	3	the	the	DET
ejpam-4835	89	4	multigrid	multigrid	NOUN
ejpam-4835	89	5	method	method	NOUN
ejpam-4835	89	6	for	for	ADP
ejpam-4835	89	7	vis	vis	X
ejpam-4835	89	8	choosing	choose	VERB
ejpam-4835	89	9	an	an	DET
ejpam-4835	89	10	iteration	iteration	NOUN
ejpam-4835	89	11	uνk	uνk	ADV
ejpam-4835	89	12	,	,	PUNCT
ejpam-4835	89	13	ν	ν	X
ejpam-4835	89	14	>	>	X
ejpam-4835	89	15	0	0	NUM
ejpam-4835	89	16	for	for	ADP
ejpam-4835	89	17	the	the	DET
ejpam-4835	89	18	multigrid	multigrid	NOUN
ejpam-4835	89	19	method	method	NOUN
ejpam-4835	89	20	,	,	PUNCT
ejpam-4835	89	21	we	we	PRON
ejpam-4835	89	22	obtain	obtain	VERB
ejpam-4835	89	23	ūνk	ūνk	PROPN
ejpam-4835	89	24	,	,	PUNCT
ejpam-4835	89	25	by	by	ADP
ejpam-4835	89	26	applying	apply	VERB
ejpam-4835	89	27	an	an	DET
ejpam-4835	89	28	iterative	iterative	NOUN
ejpam-4835	89	29	method	method	NOUN
ejpam-4835	89	30	to	to	PART
ejpam-4835	89	31	solve	solve	VERB
ejpam-4835	89	32	the	the	DET
ejpam-4835	89	33	system	system	NOUN
ejpam-4835	89	34	(	(	PUNCT
ejpam-4835	89	35	2.7	2.7	NUM
ejpam-4835	89	36	)	)	PUNCT
ejpam-4835	89	37	by	by	ADP
ejpam-4835	89	38	α	α	X
ejpam-4835	89	39	,	,	PUNCT
ejpam-4835	89	40	expressing	express	VERB
ejpam-4835	89	41	for	for	ADP
ejpam-4835	89	42	ūνk	ūνk	PROPN
ejpam-4835	89	43	=	=	PRON
ejpam-4835	90	1	sα	sα	PROPN
ejpam-4835	90	2	k	k	X
ejpam-4835	90	3	(	(	PUNCT
ejpam-4835	90	4	uνk	uνk	PROPN
ejpam-4835	90	5	)	)	PUNCT
ejpam-4835	90	6	(	(	PUNCT
ejpam-4835	90	7	2.11	2.11	NUM
ejpam-4835	90	8	)	)	PUNCT
ejpam-4835	90	9	where	where	SCONJ
ejpam-4835	90	10	sk	sk	NOUN
ejpam-4835	90	11	represents	represent	VERB
ejpam-4835	90	12	the	the	DET
ejpam-4835	90	13	iteration	iteration	NOUN
ejpam-4835	90	14	or	or	CCONJ
ejpam-4835	90	15	smoothing	smoothing	NOUN
ejpam-4835	90	16	operator	operator	NOUN
ejpam-4835	90	17	,	,	PUNCT
ejpam-4835	90	18	and	and	CCONJ
ejpam-4835	90	19	α	α	PRON
ejpam-4835	90	20	denotes	denote	VERB
ejpam-4835	90	21	the	the	DET
ejpam-4835	90	22	number	number	NOUN
ejpam-4835	90	23	of	of	ADP
ejpam-4835	90	24	iterations	iteration	NOUN
ejpam-4835	90	25	performed	perform	VERB
ejpam-4835	90	26	.	.	PUNCT
ejpam-4835	91	1	we	we	PRON
ejpam-4835	91	2	denote	denote	VERB
ejpam-4835	91	3	the	the	DET
ejpam-4835	91	4	solution	solution	NOUN
ejpam-4835	91	5	to	to	ADP
ejpam-4835	91	6	(	(	PUNCT
ejpam-4835	91	7	2.7	2.7	NUM
ejpam-4835	91	8	)	)	PUNCT
ejpam-4835	91	9	by	by	ADP
ejpam-4835	91	10	u∗k	u∗k	PROPN
ejpam-4835	91	11	.	.	PUNCT
ejpam-4835	92	1	the	the	DET
ejpam-4835	92	2	error	error	NOUN
ejpam-4835	92	3	setting	set	VERB
ejpam-4835	92	4	e	e	NOUN
ejpam-4835	92	5	ν	ν	X
ejpam-4835	92	6	k	k	X
ejpam-4835	92	7	=	=	PUNCT
ejpam-4835	92	8	ūνk−u∗k	ūνk−u∗k	X
ejpam-4835	92	9	,	,	PUNCT
ejpam-4835	92	10	and	and	CCONJ
ejpam-4835	92	11	the	the	DET
ejpam-4835	92	12	residual	residual	ADJ
ejpam-4835	92	13	d	d	X
ejpam-4835	92	14	(	(	PUNCT
ejpam-4835	92	15	ν	ν	NOUN
ejpam-4835	92	16	)	)	PUNCT
ejpam-4835	92	17	k	k	NOUN
ejpam-4835	92	18	=	=	PUNCT
ejpam-4835	92	19	zν	zν	PROPN
ejpam-4835	92	20	k	k	PROPN
ejpam-4835	92	21	−aν	−aν	PROPN
ejpam-4835	92	22	kū	kū	PROPN
ejpam-4835	92	23	ν	ν	PROPN
ejpam-4835	92	24	k	k	PROPN
ejpam-4835	92	25	,	,	PUNCT
ejpam-4835	92	26	we	we	PRON
ejpam-4835	92	27	can	can	AUX
ejpam-4835	92	28	write	write	VERB
ejpam-4835	92	29	the	the	DET
ejpam-4835	92	30	equation	equation	NOUN
ejpam-4835	92	31	(	(	PUNCT
ejpam-4835	92	32	2.7	2.7	NUM
ejpam-4835	92	33	)	)	PUNCT
ejpam-4835	92	34	as	as	ADP
ejpam-4835	92	35	aν	aν	PROPN
ejpam-4835	92	36	k	k	PROPN
ejpam-4835	92	37	(	(	PUNCT
ejpam-4835	92	38	ū	ū	NOUN
ejpam-4835	92	39	ν	ν	X
ejpam-4835	92	40	k	k	PROPN
ejpam-4835	92	41	+	+	CCONJ
ejpam-4835	92	42	eνk	eνk	NOUN
ejpam-4835	92	43	)	)	PUNCT
ejpam-4835	92	44	=	=	SYM
ejpam-4835	93	1	zν	zν	INTJ
ejpam-4835	93	2	k	k	PROPN
ejpam-4835	93	3	.	.	PUNCT
ejpam-4835	94	1	this	this	PRON
ejpam-4835	94	2	leads	lead	VERB
ejpam-4835	94	3	to	to	ADP
ejpam-4835	94	4	the	the	DET
ejpam-4835	94	5	residual	residual	ADJ
ejpam-4835	94	6	equation	equation	NOUN
ejpam-4835	94	7	aν	aν	NOUN
ejpam-4835	94	8	ke	ke	NOUN
ejpam-4835	94	9	ν	ν	X
ejpam-4835	94	10	k	k	PROPN
ejpam-4835	95	1	=	=	PUNCT
ejpam-4835	95	2	zν	zν	PROPN
ejpam-4835	95	3	k	k	PROPN
ejpam-4835	95	4	−aν	−aν	PROPN
ejpam-4835	95	5	kū	kū	PROPN
ejpam-4835	95	6	ν	ν	X
ejpam-4835	95	7	k	k	PROPN
ejpam-4835	95	8	=	=	SYM
ejpam-4835	95	9	d	d	PROPN
ejpam-4835	95	10	(	(	PUNCT
ejpam-4835	95	11	ν	ν	NOUN
ejpam-4835	95	12	)	)	PUNCT
ejpam-4835	95	13	k	k	PROPN
ejpam-4835	95	14	.	.	PUNCT
ejpam-4835	96	1	on	on	ADP
ejpam-4835	96	2	the	the	DET
ejpam-4835	96	3	fine	fine	ADJ
ejpam-4835	96	4	grid	grid	NOUN
ejpam-4835	96	5	,	,	PUNCT
ejpam-4835	96	6	after	after	ADP
ejpam-4835	96	7	relaxation	relaxation	NOUN
ejpam-4835	96	8	on	on	ADP
ejpam-4835	96	9	aν	aν	PROPN
ejpam-4835	96	10	kū	kū	PROPN
ejpam-4835	96	11	ν	ν	PROPN
ejpam-4835	96	12	k	k	PROPN
ejpam-4835	96	13	=	=	PUNCT
ejpam-4835	96	14	zν	zν	PROPN
ejpam-4835	96	15	k	k	PROPN
ejpam-4835	96	16	,	,	PUNCT
ejpam-4835	96	17	the	the	DET
ejpam-4835	96	18	error	error	NOUN
ejpam-4835	96	19	will	will	AUX
ejpam-4835	96	20	exhibit	exhibit	VERB
ejpam-4835	96	21	smooth	smooth	ADJ
ejpam-4835	96	22	behavior	behavior	NOUN
ejpam-4835	96	23	.	.	PUNCT
ejpam-4835	97	1	however	however	ADV
ejpam-4835	97	2	,	,	PUNCT
ejpam-4835	97	3	on	on	ADP
ejpam-4835	97	4	the	the	DET
ejpam-4835	97	5	coarse	coarse	ADJ
ejpam-4835	97	6	grid	grid	NOUN
ejpam-4835	97	7	,	,	PUNCT
ejpam-4835	97	8	the	the	DET
ejpam-4835	97	9	error	error	NOUN
ejpam-4835	97	10	appears	appear	VERB
ejpam-4835	97	11	to	to	PART
ejpam-4835	97	12	be	be	AUX
ejpam-4835	97	13	more	more	ADV
ejpam-4835	97	14	oscillatory	oscillatory	ADJ
ejpam-4835	97	15	,	,	PUNCT
ejpam-4835	97	16	leading	lead	VERB
ejpam-4835	97	17	to	to	ADP
ejpam-4835	97	18	efficient	efficient	ADJ
ejpam-4835	97	19	relaxation	relaxation	NOUN
ejpam-4835	97	20	.	.	PUNCT
ejpam-4835	98	1	to	to	PART
ejpam-4835	98	2	completely	completely	ADV
ejpam-4835	98	3	determine	determine	VERB
ejpam-4835	98	4	eνk	eνk	NOUN
ejpam-4835	98	5	,	,	PUNCT
ejpam-4835	98	6	we	we	PRON
ejpam-4835	98	7	need	need	VERB
ejpam-4835	98	8	to	to	PART
ejpam-4835	98	9	compute	compute	VERB
ejpam-4835	98	10	eνk−1	eνk−1	PROPN
ejpam-4835	98	11	at	at	ADP
ejpam-4835	98	12	the	the	DET
ejpam-4835	98	13	(	(	PUNCT
ejpam-4835	98	14	k	k	NOUN
ejpam-4835	98	15	−	−	PROPN
ejpam-4835	98	16	1	1	NUM
ejpam-4835	98	17	)	)	PUNCT
ejpam-4835	98	18	level	level	NOUN
ejpam-4835	98	19	,	,	PUNCT
ejpam-4835	98	20	as	as	SCONJ
ejpam-4835	98	21	it	it	PRON
ejpam-4835	98	22	serves	serve	VERB
ejpam-4835	98	23	as	as	ADP
ejpam-4835	98	24	the	the	DET
ejpam-4835	98	25	solution	solution	NOUN
ejpam-4835	98	26	for	for	ADP
ejpam-4835	98	27	the	the	DET
ejpam-4835	98	28	coarse	coarse	ADJ
ejpam-4835	98	29	grid	grid	NOUN
ejpam-4835	98	30	system	system	NOUN
ejpam-4835	98	31	.	.	PUNCT
ejpam-4835	99	1	aν	aν	NOUN
ejpam-4835	99	2	k−1e	k−1e	PROPN
ejpam-4835	100	1	ν	ν	X
ejpam-4835	100	2	k−1	k−1	PROPN
ejpam-4835	100	3	=	=	SYM
ejpam-4835	100	4	d	d	PROPN
ejpam-4835	100	5	(	(	PUNCT
ejpam-4835	100	6	ν	ν	NOUN
ejpam-4835	100	7	)	)	PUNCT
ejpam-4835	100	8	k−1	k−1	PROPN
ejpam-4835	100	9	.	.	PUNCT
ejpam-4835	101	1	(	(	PUNCT
ejpam-4835	101	2	2.12	2.12	NUM
ejpam-4835	101	3	)	)	PUNCT
ejpam-4835	101	4	h.	h.	PROPN
ejpam-4835	101	5	ahmad	ahmad	PROPN
ejpam-4835	101	6	et	et	PROPN
ejpam-4835	101	7	al	al	PROPN
ejpam-4835	101	8	.	.	PUNCT
ejpam-4835	101	9	/	/	SYM
ejpam-4835	101	10	eur	eur	PROPN
ejpam-4835	101	11	.	.	PUNCT
ejpam-4835	102	1	j.	j.	PROPN
ejpam-4835	102	2	pure	pure	PROPN
ejpam-4835	102	3	appl	appl	PROPN
ejpam-4835	102	4	.	.	PROPN
ejpam-4835	102	5	math	math	PROPN
ejpam-4835	102	6	,	,	PUNCT
ejpam-4835	102	7	16	16	NUM
ejpam-4835	102	8	(	(	PUNCT
ejpam-4835	102	9	3	3	NUM
ejpam-4835	102	10	)	)	PUNCT
ejpam-4835	102	11	(	(	PUNCT
ejpam-4835	102	12	2023	2023	NUM
ejpam-4835	102	13	)	)	PUNCT
ejpam-4835	102	14	,	,	PUNCT
ejpam-4835	102	15	1956	1956	NUM
ejpam-4835	102	16	-	-	SYM
ejpam-4835	102	17	1969	1969	NUM
ejpam-4835	102	18	1961	1961	NUM
ejpam-4835	102	19	we	we	PRON
ejpam-4835	102	20	can	can	AUX
ejpam-4835	102	21	interpret	interpret	VERB
ejpam-4835	102	22	eνk−1	eνk−1	PROPN
ejpam-4835	102	23	(	(	PUNCT
ejpam-4835	102	24	and	and	CCONJ
ejpam-4835	102	25	its	its	PRON
ejpam-4835	102	26	corresponding	correspond	VERB
ejpam-4835	102	27	operators	operator	NOUN
ejpam-4835	102	28	aν	aν	PUNCT
ejpam-4835	102	29	k−1	k−1	PROPN
ejpam-4835	102	30	,	,	PUNCT
ejpam-4835	102	31	d	d	X
ejpam-4835	102	32	(	(	PUNCT
ejpam-4835	102	33	ν	ν	NOUN
ejpam-4835	102	34	)	)	PUNCT
ejpam-4835	102	35	k−1	k−1	PROPN
ejpam-4835	102	36	)	)	PUNCT
ejpam-4835	102	37	as	as	ADP
ejpam-4835	102	38	the	the	DET
ejpam-4835	102	39	approximation	approximation	NOUN
ejpam-4835	102	40	operator	operator	NOUN
ejpam-4835	102	41	at	at	ADP
ejpam-4835	102	42	level	level	NOUN
ejpam-4835	102	43	k	k	NOUN
ejpam-4835	102	44	−	−	PROPN
ejpam-4835	102	45	1	1	NUM
ejpam-4835	102	46	,	,	PUNCT
ejpam-4835	102	47	while	while	SCONJ
ejpam-4835	102	48	eνk	eνk	NOUN
ejpam-4835	102	49	(	(	PUNCT
ejpam-4835	102	50	and	and	CCONJ
ejpam-4835	102	51	its	its	PRON
ejpam-4835	102	52	corresponding	correspond	VERB
ejpam-4835	102	53	operators	operator	NOUN
ejpam-4835	102	54	aν	aν	VERB
ejpam-4835	103	1	k	k	NOUN
ejpam-4835	103	2	,	,	PUNCT
ejpam-4835	103	3	d	d	X
ejpam-4835	103	4	(	(	PUNCT
ejpam-4835	103	5	ν	ν	NOUN
ejpam-4835	103	6	)	)	PUNCT
ejpam-4835	103	7	k	k	NOUN
ejpam-4835	103	8	)	)	PUNCT
ejpam-4835	103	9	represent	represent	VERB
ejpam-4835	103	10	the	the	DET
ejpam-4835	103	11	approximation	approximation	NOUN
ejpam-4835	103	12	operator	operator	NOUN
ejpam-4835	103	13	at	at	ADP
ejpam-4835	103	14	level	level	NOUN
ejpam-4835	103	15	k.	k.	PROPN
ejpam-4835	104	1	additionally	additionally	ADV
ejpam-4835	104	2	,	,	PUNCT
ejpam-4835	104	3	we	we	PRON
ejpam-4835	104	4	have	have	VERB
ejpam-4835	104	5	the	the	DET
ejpam-4835	104	6	restriction	restriction	NOUN
ejpam-4835	104	7	operator	operator	NOUN
ejpam-4835	104	8	rk	rk	NOUN
ejpam-4835	104	9	and	and	CCONJ
ejpam-4835	104	10	its	its	PRON
ejpam-4835	104	11	inverse	inverse	NOUN
ejpam-4835	104	12	pk	pk	NOUN
ejpam-4835	104	13	.	.	PUNCT
ejpam-4835	105	1	therefore	therefore	ADV
ejpam-4835	105	2	,	,	PUNCT
ejpam-4835	105	3	we	we	PRON
ejpam-4835	105	4	identify	identify	VERB
ejpam-4835	105	5	an	an	DET
ejpam-4835	105	6	improved	improved	ADJ
ejpam-4835	105	7	iteration	iteration	NOUN
ejpam-4835	105	8	at	at	ADP
ejpam-4835	105	9	the	the	DET
ejpam-4835	105	10	k	k	PROPN
ejpam-4835	105	11	level	level	NOUN
ejpam-4835	105	12	uν+1	uν+1	PROPN
ejpam-4835	105	13	k	k	PROPN
ejpam-4835	105	14	=	=	PUNCT
ejpam-4835	105	15	ūνk	ūνk	PROPN
ejpam-4835	105	16	+	+	CCONJ
ejpam-4835	105	17	pk	pk	PROPN
ejpam-4835	105	18	(	(	PUNCT
ejpam-4835	105	19	eνk−1	eνk−1	PROPN
ejpam-4835	105	20	)	)	PUNCT
ejpam-4835	105	21	.	.	PUNCT
ejpam-4835	106	1	(	(	PUNCT
ejpam-4835	106	2	2.13	2.13	NUM
ejpam-4835	106	3	)	)	PUNCT
ejpam-4835	106	4	due	due	ADP
ejpam-4835	106	5	to	to	ADP
ejpam-4835	106	6	the	the	DET
ejpam-4835	106	7	nestedness	nestedness	NOUN
ejpam-4835	106	8	,	,	PUNCT
ejpam-4835	106	9	we	we	PRON
ejpam-4835	106	10	use	use	VERB
ejpam-4835	106	11	the	the	DET
ejpam-4835	106	12	well	well	ADV
ejpam-4835	106	13	-	-	PUNCT
ejpam-4835	106	14	defined	define	VERB
ejpam-4835	106	15	identity	identity	NOUN
ejpam-4835	106	16	operator	operator	NOUN
ejpam-4835	106	17	π	π	NOUN
ejpam-4835	106	18	:	:	PUNCT
ejpam-4835	106	19	vk−1	vk−1	VERB
ejpam-4835	106	20	−→	−→	ADJ
ejpam-4835	106	21	vk	vk	NOUN
ejpam-4835	106	22	,	,	PUNCT
ejpam-4835	106	23	πv	πv	NOUN
ejpam-4835	106	24	=	=	SYM
ejpam-4835	106	25	v	v	NOUN
ejpam-4835	106	26	,	,	PUNCT
ejpam-4835	106	27	to	to	PART
ejpam-4835	106	28	define	define	VERB
ejpam-4835	106	29	prolongation	prolongation	NOUN
ejpam-4835	106	30	and	and	CCONJ
ejpam-4835	106	31	restriction	restriction	NOUN
ejpam-4835	106	32	operators	operator	NOUN
ejpam-4835	106	33	,	,	PUNCT
ejpam-4835	106	34	that	that	PRON
ejpam-4835	106	35	is	be	AUX
ejpam-4835	106	36	:	:	PUNCT
ejpam-4835	106	37	pk	pk	NOUN
ejpam-4835	106	38	=	=	NOUN
ejpam-4835	106	39	r−1	r−1	PROPN
ejpam-4835	106	40	k	k	PROPN
ejpam-4835	106	41	rk−1	rk−1	PROPN
ejpam-4835	106	42	,	,	PUNCT
ejpam-4835	106	43	rk	rk	NOUN
ejpam-4835	106	44	=	=	SYM
ejpam-4835	106	45	pt	pt	PROPN
ejpam-4835	106	46	k.	k.	PROPN
ejpam-4835	106	47	(	(	PUNCT
ejpam-4835	106	48	2.14	2.14	NUM
ejpam-4835	106	49	)	)	PUNCT
ejpam-4835	106	50	note	note	VERB
ejpam-4835	106	51	4	4	NUM
ejpam-4835	106	52	:	:	PUNCT
ejpam-4835	106	53	the	the	DET
ejpam-4835	106	54	preceding	precede	VERB
ejpam-4835	106	55	algorithm	algorithm	NOUN
ejpam-4835	106	56	describes	describe	VERB
ejpam-4835	106	57	a	a	DET
ejpam-4835	106	58	loop	loop	NOUN
ejpam-4835	106	59	of	of	ADP
ejpam-4835	106	60	two	two	NUM
ejpam-4835	106	61	mesh	mesh	NOUN
ejpam-4835	106	62	iterations	iteration	NOUN
ejpam-4835	106	63	to	to	PART
ejpam-4835	106	64	solve	solve	VERB
ejpam-4835	106	65	(	(	PUNCT
ejpam-4835	106	66	2.7	2.7	NUM
ejpam-4835	106	67	)	)	PUNCT
ejpam-4835	106	68	for	for	ADP
ejpam-4835	106	69	two	two	NUM
ejpam-4835	106	70	mesh	mesh	NOUN
ejpam-4835	106	71	levels	level	NOUN
ejpam-4835	106	72	ωk−1	ωk−1	VERB
ejpam-4835	106	73	.	.	PUNCT
ejpam-4835	107	1	it	it	PRON
ejpam-4835	107	2	is	be	AUX
ejpam-4835	107	3	clear	clear	ADJ
ejpam-4835	107	4	that	that	SCONJ
ejpam-4835	107	5	the	the	DET
ejpam-4835	107	6	coarse	coarse	ADJ
ejpam-4835	107	7	grid	grid	NOUN
ejpam-4835	107	8	system	system	NOUN
ejpam-4835	107	9	(	(	PUNCT
ejpam-4835	107	10	2.12	2.12	NUM
ejpam-4835	107	11	)	)	PUNCT
ejpam-4835	107	12	has	have	VERB
ejpam-4835	107	13	the	the	DET
ejpam-4835	107	14	same	same	ADJ
ejpam-4835	107	15	form	form	NOUN
ejpam-4835	107	16	as	as	ADP
ejpam-4835	107	17	the	the	DET
ejpam-4835	107	18	system	system	NOUN
ejpam-4835	107	19	(	(	PUNCT
ejpam-4835	107	20	2.7	2.7	NUM
ejpam-4835	107	21	)	)	PUNCT
ejpam-4835	107	22	.	.	PUNCT
ejpam-4835	108	1	therefore	therefore	ADV
ejpam-4835	108	2	,	,	PUNCT
ejpam-4835	108	3	we	we	PRON
ejpam-4835	108	4	can	can	AUX
ejpam-4835	108	5	approximate	approximate	VERB
ejpam-4835	108	6	the	the	DET
ejpam-4835	108	7	solution	solution	NOUN
ejpam-4835	108	8	of	of	ADP
ejpam-4835	108	9	the	the	DET
ejpam-4835	108	10	system	system	NOUN
ejpam-4835	108	11	(	(	PUNCT
ejpam-4835	108	12	2.12	2.12	NUM
ejpam-4835	108	13	)	)	PUNCT
ejpam-4835	108	14	by	by	ADP
ejpam-4835	108	15	recursively	recursively	ADV
ejpam-4835	108	16	doing	do	VERB
ejpam-4835	108	17	two	two	NUM
ejpam-4835	108	18	-	-	PUNCT
ejpam-4835	108	19	grid	grid	NOUN
ejpam-4835	108	20	iterations	iteration	NOUN
ejpam-4835	108	21	at	at	ADP
ejpam-4835	108	22	all	all	DET
ejpam-4835	108	23	grid	grid	NOUN
ejpam-4835	108	24	levels	level	NOUN
ejpam-4835	108	25	{	{	PUNCT
ejpam-4835	108	26	ωk	ωk	ADP
ejpam-4835	108	27	,	,	PUNCT
ejpam-4835	108	28	k	k	NOUN
ejpam-4835	108	29	=	=	PUNCT
ejpam-4835	108	30	0	0	PROPN
ejpam-4835	108	31	,	,	PUNCT
ejpam-4835	108	32	.	.	PUNCT
ejpam-4835	108	33	.	.	PUNCT
ejpam-4835	109	1	.	.	PUNCT
ejpam-4835	110	1	,	,	PUNCT
ejpam-4835	110	2	mk	mk	PROPN
ejpam-4835	110	3	}	}	PUNCT
ejpam-4835	110	4	.	.	PUNCT
ejpam-4835	111	1	2.7	2.7	NUM
ejpam-4835	111	2	.	.	PUNCT
ejpam-4835	111	3	matrix	matrix	NOUN
ejpam-4835	111	4	associated	associate	VERB
ejpam-4835	111	5	with	with	ADP
ejpam-4835	111	6	the	the	DET
ejpam-4835	111	7	mghjb	mghjb	NOUN
ejpam-4835	111	8	algorithm	algorithm	PROPN
ejpam-4835	111	9	:	:	PUNCT
ejpam-4835	111	10	the	the	DET
ejpam-4835	111	11	matrix	matrix	NOUN
ejpam-4835	111	12	of	of	ADP
ejpam-4835	111	13	iterations	iteration	NOUN
ejpam-4835	111	14	for	for	ADP
ejpam-4835	111	15	the	the	DET
ejpam-4835	111	16	two	two	NUM
ejpam-4835	111	17	-	-	PUNCT
ejpam-4835	111	18	grid	grid	NOUN
ejpam-4835	111	19	method	method	NOUN
ejpam-4835	111	20	with	with	ADP
ejpam-4835	111	21	α1	α1	PROPN
ejpam-4835	111	22	presmoothing	presmoothing	NOUN
ejpam-4835	111	23	and	and	CCONJ
ejpam-4835	111	24	α2	α2	ADJ
ejpam-4835	111	25	postsmoothing	postsmoothe	VERB
ejpam-4835	111	26	at	at	ADP
ejpam-4835	111	27	level	level	NOUN
ejpam-4835	111	28	k	k	PROPN
ejpam-4835	111	29	is	be	AUX
ejpam-4835	111	30	expressed	express	VERB
ejpam-4835	111	31	as	as	SCONJ
ejpam-4835	111	32	follows	follow	VERB
ejpam-4835	111	33	:	:	PUNCT
ejpam-4835	111	34	tgk	tgk	PROPN
ejpam-4835	111	35	(	(	PUNCT
ejpam-4835	111	36	α1	α1	PROPN
ejpam-4835	111	37	,	,	PUNCT
ejpam-4835	111	38	α2	α2	ADJ
ejpam-4835	111	39	)	)	PUNCT
ejpam-4835	111	40	=	=	PUNCT
ejpam-4835	112	1	sα2	sα2	X
ejpam-4835	112	2	k	k	X
ejpam-4835	112	3	(	(	PUNCT
ejpam-4835	112	4	(	(	PUNCT
ejpam-4835	112	5	aν	aν	NOUN
ejpam-4835	112	6	k	k	NOUN
ejpam-4835	112	7	)	)	PUNCT
ejpam-4835	112	8	−1	−1	NOUN
ejpam-4835	112	9	−	−	PROPN
ejpam-4835	112	10	pk	pk	NOUN
ejpam-4835	112	11	(	(	PUNCT
ejpam-4835	112	12	aν	aν	NOUN
ejpam-4835	112	13	k−1	k−1	PROPN
ejpam-4835	112	14	)	)	PUNCT
ejpam-4835	112	15	−1rk	−1rk	NOUN
ejpam-4835	112	16	)	)	PUNCT
ejpam-4835	112	17	(	(	PUNCT
ejpam-4835	112	18	aν	aν	NOUN
ejpam-4835	112	19	k)s	k)s	NOUN
ejpam-4835	112	20	α1	α1	PROPN
ejpam-4835	112	21	k	k	X
ejpam-4835	112	22	.	.	PUNCT
ejpam-4835	113	1	(	(	PUNCT
ejpam-4835	113	2	2.15	2.15	NUM
ejpam-4835	113	3	)	)	PUNCT
ejpam-4835	113	4	theorem	theorem	NOUN
ejpam-4835	113	5	5	5	NUM
ejpam-4835	113	6	:	:	PUNCT
ejpam-4835	113	7	(	(	PUNCT
ejpam-4835	113	8	[	[	X
ejpam-4835	113	9	26	26	NUM
ejpam-4835	113	10	]	]	PUNCT
ejpam-4835	113	11	)	)	PUNCT
ejpam-4835	113	12	the	the	DET
ejpam-4835	113	13	multigrid	multigrid	PROPN
ejpam-4835	113	14	approach	approach	NOUN
ejpam-4835	113	15	constitutes	constitute	VERB
ejpam-4835	113	16	a	a	DET
ejpam-4835	113	17	linear	linear	ADJ
ejpam-4835	113	18	iterative	iterative	NOUN
ejpam-4835	113	19	method	method	NOUN
ejpam-4835	113	20	,	,	PUNCT
ejpam-4835	113	21	with	with	ADP
ejpam-4835	113	22	the	the	DET
ejpam-4835	113	23	iteration	iteration	NOUN
ejpam-4835	113	24	matrix	matrix	NOUN
ejpam-4835	113	25	denoted	denote	VERB
ejpam-4835	113	26	as	as	ADP
ejpam-4835	113	27	mgk	mgk	PROPN
ejpam-4835	113	28	.	.	PUNCT
ejpam-4835	113	29	mg0	mg0	PROPN
ejpam-4835	113	30	=	=	SYM
ejpam-4835	113	31	0	0	PROPN
ejpam-4835	113	32	,	,	PUNCT
ejpam-4835	113	33	mgk	mgk	NOUN
ejpam-4835	113	34	=	=	SYM
ejpam-4835	113	35	sα2	sα2	X
ejpam-4835	113	36	k	k	X
ejpam-4835	113	37	(	(	PUNCT
ejpam-4835	113	38	ik	ik	PROPN
ejpam-4835	113	39	−	−	PROPN
ejpam-4835	113	40	pk	pk	PROPN
ejpam-4835	113	41	(	(	PUNCT
ejpam-4835	113	42	ik	ik	PROPN
ejpam-4835	113	43	−mgk−1	−mgk−1	PROPN
ejpam-4835	113	44	)	)	PUNCT
ejpam-4835	114	1	(	(	PUNCT
ejpam-4835	114	2	aν	aν	NOUN
ejpam-4835	114	3	k−1	k−1	PROPN
ejpam-4835	114	4	)	)	PUNCT
ejpam-4835	114	5	−1rk	−1rk	NOUN
ejpam-4835	114	6	)	)	PUNCT
ejpam-4835	115	1	(	(	PUNCT
ejpam-4835	115	2	aν	aν	NOUN
ejpam-4835	115	3	k)s	k)s	X
ejpam-4835	115	4	α1	α1	PROPN
ejpam-4835	115	5	k	k	PROPN
ejpam-4835	115	6	,	,	PUNCT
ejpam-4835	115	7	=	=	SYM
ejpam-4835	115	8	tgk	tgk	PROPN
ejpam-4835	115	9	+	+	CCONJ
ejpam-4835	115	10	sα2	sα2	X
ejpam-4835	115	11	k	k	X
ejpam-4835	115	12	pkmgk−1	pkmgk−1	NOUN
ejpam-4835	115	13	(	(	PUNCT
ejpam-4835	115	14	aν	aν	NOUN
ejpam-4835	115	15	k−1	k−1	PROPN
ejpam-4835	115	16	)	)	PUNCT
ejpam-4835	116	1	−1rk	−1rk	NOUN
ejpam-4835	116	2	(	(	PUNCT
ejpam-4835	116	3	aν	aν	NOUN
ejpam-4835	116	4	k)s	k)s	NOUN
ejpam-4835	116	5	α1	α1	PROPN
ejpam-4835	117	1	k	k	PROPN
ejpam-4835	117	2	,	,	PUNCT
ejpam-4835	117	3	k	k	PROPN
ejpam-4835	117	4	=	=	SYM
ejpam-4835	117	5	1	1	NUM
ejpam-4835	117	6	,	,	PUNCT
ejpam-4835	117	7	2	2	NUM
ejpam-4835	117	8	,	,	PUNCT
ejpam-4835	117	9	..	..	PUNCT
ejpam-4835	117	10	(	(	PUNCT
ejpam-4835	117	11	2.16	2.16	NUM
ejpam-4835	117	12	)	)	PUNCT
ejpam-4835	117	13	3	3	NUM
ejpam-4835	117	14	.	.	X
ejpam-4835	117	15	convergence	convergence	NOUN
ejpam-4835	117	16	analysis	analysis	NOUN
ejpam-4835	117	17	of	of	ADP
ejpam-4835	117	18	multigrid	multigrid	NOUN
ejpam-4835	117	19	method	method	NOUN
ejpam-4835	117	20	in	in	ADP
ejpam-4835	117	21	l∞	l∞	NOUN
ejpam-4835	117	22	norm	norm	NOUN
ejpam-4835	117	23	in	in	ADP
ejpam-4835	117	24	this	this	DET
ejpam-4835	117	25	section	section	NOUN
ejpam-4835	117	26	,	,	PUNCT
ejpam-4835	117	27	we	we	PRON
ejpam-4835	117	28	provide	provide	VERB
ejpam-4835	117	29	a	a	DET
ejpam-4835	117	30	comprehensive	comprehensive	ADJ
ejpam-4835	117	31	convergence	convergence	NOUN
ejpam-4835	117	32	analysis	analysis	NOUN
ejpam-4835	117	33	of	of	ADP
ejpam-4835	117	34	the	the	DET
ejpam-4835	117	35	multigrid	multigrid	NOUN
ejpam-4835	117	36	algorithm	algorithm	NOUN
ejpam-4835	117	37	.	.	PUNCT
ejpam-4835	118	1	the	the	DET
ejpam-4835	118	2	algorithm	algorithm	NOUN
ejpam-4835	118	3	was	be	AUX
ejpam-4835	118	4	previously	previously	ADV
ejpam-4835	118	5	described	describe	VERB
ejpam-4835	118	6	using	use	VERB
ejpam-4835	118	7	the	the	DET
ejpam-4835	118	8	maximum	maximum	ADJ
ejpam-4835	118	9	norm	norm	NOUN
ejpam-4835	118	10	in	in	ADP
ejpam-4835	118	11	the	the	DET
ejpam-4835	118	12	preceding	precede	VERB
ejpam-4835	118	13	section	section	NOUN
ejpam-4835	118	14	.	.	PUNCT
ejpam-4835	119	1	h.	h.	PROPN
ejpam-4835	119	2	ahmad	ahmad	PROPN
ejpam-4835	119	3	et	et	PROPN
ejpam-4835	119	4	al	al	PROPN
ejpam-4835	119	5	.	.	PUNCT
ejpam-4835	119	6	/	/	SYM
ejpam-4835	119	7	eur	eur	PROPN
ejpam-4835	119	8	.	.	PUNCT
ejpam-4835	120	1	j.	j.	PROPN
ejpam-4835	120	2	pure	pure	PROPN
ejpam-4835	120	3	appl	appl	PROPN
ejpam-4835	120	4	.	.	PROPN
ejpam-4835	120	5	math	math	PROPN
ejpam-4835	120	6	,	,	PUNCT
ejpam-4835	120	7	16	16	NUM
ejpam-4835	120	8	(	(	PUNCT
ejpam-4835	120	9	3	3	NUM
ejpam-4835	120	10	)	)	PUNCT
ejpam-4835	120	11	(	(	PUNCT
ejpam-4835	120	12	2023	2023	NUM
ejpam-4835	120	13	)	)	PUNCT
ejpam-4835	120	14	,	,	PUNCT
ejpam-4835	120	15	1956	1956	NUM
ejpam-4835	120	16	-	-	SYM
ejpam-4835	120	17	1969	1969	NUM
ejpam-4835	120	18	1962	1962	NUM
ejpam-4835	120	19	3.1	3.1	NUM
ejpam-4835	120	20	.	.	PUNCT
ejpam-4835	121	1	approximation	approximation	NOUN
ejpam-4835	121	2	property	property	NOUN
ejpam-4835	121	3	theorem	theorem	VERB
ejpam-4835	121	4	6	6	NUM
ejpam-4835	121	5	:	:	PUNCT
ejpam-4835	121	6	(	(	PUNCT
ejpam-4835	121	7	[	[	X
ejpam-4835	121	8	17	17	NUM
ejpam-4835	121	9	]	]	PUNCT
ejpam-4835	121	10	)	)	PUNCT
ejpam-4835	121	11	the	the	DET
ejpam-4835	121	12	matrix	matrix	NOUN
ejpam-4835	121	13	υk	υk	NOUN
ejpam-4835	121	14	=	=	SYM
ejpam-4835	121	15	[	[	PUNCT
ejpam-4835	121	16	(	(	PUNCT
ejpam-4835	121	17	aν	aν	NOUN
ejpam-4835	121	18	k	k	NOUN
ejpam-4835	121	19	)	)	PUNCT
ejpam-4835	121	20	−1	−1	NOUN
ejpam-4835	121	21	−	−	PROPN
ejpam-4835	121	22	pk	pk	NOUN
ejpam-4835	121	23	(	(	PUNCT
ejpam-4835	121	24	aν	aν	NOUN
ejpam-4835	121	25	k−1	k−1	PROPN
ejpam-4835	121	26	)	)	PUNCT
ejpam-4835	121	27	−1rk	−1rk	NOUN
ejpam-4835	121	28	]	]	PUNCT
ejpam-4835	121	29	has	have	VERB
ejpam-4835	121	30	the	the	DET
ejpam-4835	121	31	approximation	approximation	NOUN
ejpam-4835	121	32	property	property	NOUN
ejpam-4835	121	33	∥υk∥∞	∥υk∥∞	VERB
ejpam-4835	121	34	≤	≤	ADJ
ejpam-4835	121	35	ch2k	ch2k	PROPN
ejpam-4835	121	36	|lnhk|	|lnhk|	ADP
ejpam-4835	121	37	2	2	NUM
ejpam-4835	121	38	.	.	PUNCT
ejpam-4835	122	1	(	(	PUNCT
ejpam-4835	122	2	3.1	3.1	NUM
ejpam-4835	122	3	)	)	PUNCT
ejpam-4835	122	4	proof	proof	NOUN
ejpam-4835	122	5	.	.	PUNCT
ejpam-4835	123	1	the	the	DET
ejpam-4835	123	2	proof	proof	NOUN
ejpam-4835	123	3	of	of	ADP
ejpam-4835	123	4	the	the	DET
ejpam-4835	123	5	approximation	approximation	NOUN
ejpam-4835	123	6	property	property	NOUN
ejpam-4835	123	7	was	be	AUX
ejpam-4835	123	8	introduced	introduce	VERB
ejpam-4835	123	9	by	by	ADP
ejpam-4835	123	10	arnold	arnold	PROPN
ejpam-4835	123	11	in	in	ADP
ejpam-4835	123	12	[	[	X
ejpam-4835	123	13	25	25	NUM
ejpam-4835	123	14	]	]	PUNCT
ejpam-4835	123	15	,	,	PUNCT
ejpam-4835	123	16	relying	rely	VERB
ejpam-4835	123	17	on	on	ADP
ejpam-4835	123	18	theorem	theorem	ADJ
ejpam-4835	123	19	(	(	PUNCT
ejpam-4835	123	20	2	2	NUM
ejpam-4835	123	21	)	)	PUNCT
ejpam-4835	123	22	.	.	PUNCT
ejpam-4835	124	1	3.2	3.2	NUM
ejpam-4835	124	2	.	.	PUNCT
ejpam-4835	124	3	smoothing	smooth	VERB
ejpam-4835	124	4	property	property	NOUN
ejpam-4835	124	5	to	to	PART
ejpam-4835	124	6	demonstrate	demonstrate	VERB
ejpam-4835	124	7	the	the	DET
ejpam-4835	124	8	smoothness	smoothness	ADJ
ejpam-4835	124	9	property	property	NOUN
ejpam-4835	124	10	,	,	PUNCT
ejpam-4835	124	11	we	we	PRON
ejpam-4835	124	12	decompose	decompose	VERB
ejpam-4835	124	13	aν	aν	NOUN
ejpam-4835	125	1	k	k	NOUN
ejpam-4835	125	2	=	=	X
ejpam-4835	125	3	ek	ek	PROPN
ejpam-4835	125	4	−	−	PROPN
ejpam-4835	125	5	nk	nk	PROPN
ejpam-4835	125	6	and	and	CCONJ
ejpam-4835	125	7	use	use	VERB
ejpam-4835	125	8	the	the	DET
ejpam-4835	125	9	following	follow	VERB
ejpam-4835	125	10	assumptions	assumption	NOUN
ejpam-4835	125	11	:	:	PUNCT
ejpam-4835	125	12	ek	ek	PROPN
ejpam-4835	125	13	is	be	AUX
ejpam-4835	125	14	regular	regular	ADJ
ejpam-4835	125	15	and	and	CCONJ
ejpam-4835	125	16	∥∥e−1	∥∥e−1	NOUN
ejpam-4835	126	1	k	k	PROPN
ejpam-4835	126	2	nk	nk	PROPN
ejpam-4835	126	3	∥∥	∥∥	PROPN
ejpam-4835	126	4	∞	∞	PROPN
ejpam-4835	126	5	≤	≤	NUM
ejpam-4835	126	6	1	1	NUM
ejpam-4835	126	7	,	,	PUNCT
ejpam-4835	126	8	for	for	ADP
ejpam-4835	126	9	all	all	DET
ejpam-4835	126	10	k.	k.	PROPN
ejpam-4835	126	11	(	(	PUNCT
ejpam-4835	126	12	3.2	3.2	NUM
ejpam-4835	126	13	)	)	PUNCT
ejpam-4835	126	14	∥ek∥∞	∥ek∥∞	PROPN
ejpam-4835	126	15	≤	≤	PROPN
ejpam-4835	126	16	ch−2	ch−2	PROPN
ejpam-4835	126	17	k	k	PROPN
ejpam-4835	126	18	,	,	PUNCT
ejpam-4835	126	19	for	for	ADP
ejpam-4835	126	20	all	all	DET
ejpam-4835	126	21	k	k	PROPN
ejpam-4835	126	22	,	,	PUNCT
ejpam-4835	126	23	with	with	ADP
ejpam-4835	126	24	c	c	PROPN
ejpam-4835	126	25	independent	independent	ADJ
ejpam-4835	126	26	of	of	ADP
ejpam-4835	126	27	k.	k.	PROPN
ejpam-4835	126	28	(	(	PUNCT
ejpam-4835	126	29	3.3	3.3	NUM
ejpam-4835	126	30	)	)	PUNCT
ejpam-4835	126	31	for	for	ADP
ejpam-4835	126	32	the	the	DET
ejpam-4835	126	33	smoothing	smoothing	NOUN
ejpam-4835	126	34	process	process	NOUN
ejpam-4835	126	35	,	,	PUNCT
ejpam-4835	126	36	we	we	PRON
ejpam-4835	126	37	employ	employ	VERB
ejpam-4835	126	38	a	a	DET
ejpam-4835	126	39	relaxation	relaxation	NOUN
ejpam-4835	126	40	method	method	NOUN
ejpam-4835	126	41	with	with	ADP
ejpam-4835	126	42	an	an	DET
ejpam-4835	126	43	iterative	iterative	NOUN
ejpam-4835	126	44	matrix	matrix	NOUN
ejpam-4835	126	45	.	.	PUNCT
ejpam-4835	127	1	sk	sk	VERB
ejpam-4835	127	2	=	=	PUNCT
ejpam-4835	127	3	ik	ik	PROPN
ejpam-4835	127	4	−	−	PROPN
ejpam-4835	127	5	ωe−1	ωe−1	PROPN
ejpam-4835	127	6	k	k	PROPN
ejpam-4835	127	7	nk	nk	PROPN
ejpam-4835	127	8	,	,	PUNCT
ejpam-4835	127	9	ω	ω	PROPN
ejpam-4835	127	10	∈	∈	PROPN
ejpam-4835	127	11	(	(	PUNCT
ejpam-4835	127	12	0	0	NUM
ejpam-4835	127	13	,	,	PUNCT
ejpam-4835	127	14	1	1	NUM
ejpam-4835	127	15	)	)	PUNCT
ejpam-4835	127	16	.	.	PUNCT
ejpam-4835	128	1	in	in	ADP
ejpam-4835	128	2	the	the	DET
ejpam-4835	128	3	following	following	NOUN
ejpam-4835	128	4	theorem	theorem	PROPN
ejpam-4835	128	5	,	,	PUNCT
ejpam-4835	128	6	arnold	arnold	PROPN
ejpam-4835	128	7	reusken	reusken	PROPN
ejpam-4835	128	8	applied	apply	VERB
ejpam-4835	128	9	a	a	DET
ejpam-4835	128	10	concept	concept	NOUN
ejpam-4835	128	11	that	that	PRON
ejpam-4835	128	12	is	be	AUX
ejpam-4835	128	13	relevant	relevant	ADJ
ejpam-4835	128	14	to	to	ADP
ejpam-4835	128	15	our	our	PRON
ejpam-4835	128	16	work	work	NOUN
ejpam-4835	128	17	.	.	PUNCT
ejpam-4835	129	1	the	the	DET
ejpam-4835	129	2	theorem	theorem	VERB
ejpam-4835	129	3	’s	’s	PART
ejpam-4835	129	4	application	application	NOUN
ejpam-4835	129	5	is	be	AUX
ejpam-4835	129	6	appropriate	appropriate	ADJ
ejpam-4835	129	7	as	as	SCONJ
ejpam-4835	129	8	it	it	PRON
ejpam-4835	129	9	is	be	AUX
ejpam-4835	129	10	closely	closely	ADV
ejpam-4835	129	11	linked	link	VERB
ejpam-4835	129	12	to	to	ADP
ejpam-4835	129	13	the	the	DET
ejpam-4835	129	14	operator	operator	NOUN
ejpam-4835	129	15	used	use	VERB
ejpam-4835	129	16	in	in	ADP
ejpam-4835	129	17	our	our	PRON
ejpam-4835	129	18	study	study	NOUN
ejpam-4835	129	19	.	.	PUNCT
ejpam-4835	130	1	theorem	theorem	VERB
ejpam-4835	130	2	7	7	NUM
ejpam-4835	130	3	:	:	PUNCT
ejpam-4835	130	4	(	(	PUNCT
ejpam-4835	130	5	[	[	X
ejpam-4835	130	6	25	25	NUM
ejpam-4835	130	7	]	]	PUNCT
ejpam-4835	130	8	)	)	PUNCT
ejpam-4835	130	9	under	under	ADP
ejpam-4835	130	10	the	the	DET
ejpam-4835	130	11	given	give	VERB
ejpam-4835	130	12	assumptions	assumption	NOUN
ejpam-4835	130	13	and	and	CCONJ
ejpam-4835	130	14	notations	notation	NOUN
ejpam-4835	130	15	,	,	PUNCT
ejpam-4835	130	16	it	it	PRON
ejpam-4835	130	17	can	can	AUX
ejpam-4835	130	18	be	be	AUX
ejpam-4835	130	19	inferred	infer	VERB
ejpam-4835	130	20	that	that	SCONJ
ejpam-4835	130	21	there	there	PRON
ejpam-4835	130	22	exists	exist	VERB
ejpam-4835	130	23	a	a	DET
ejpam-4835	130	24	constant	constant	ADJ
ejpam-4835	130	25	c	c	NOUN
ejpam-4835	130	26	,	,	PUNCT
ejpam-4835	130	27	which	which	PRON
ejpam-4835	130	28	is	be	AUX
ejpam-4835	130	29	independent	independent	ADJ
ejpam-4835	130	30	of	of	ADP
ejpam-4835	130	31	both	both	DET
ejpam-4835	130	32	k	k	PROPN
ejpam-4835	130	33	and	and	CCONJ
ejpam-4835	130	34	α	α	NOUN
ejpam-4835	130	35	.	.	PUNCT
ejpam-4835	131	1	with	with	ADP
ejpam-4835	131	2	this	this	DET
ejpam-4835	131	3	constant	constant	ADJ
ejpam-4835	131	4	,	,	PUNCT
ejpam-4835	131	5	the	the	DET
ejpam-4835	131	6	following	follow	VERB
ejpam-4835	131	7	smoothness	smoothness	ADJ
ejpam-4835	131	8	properties	property	NOUN
ejpam-4835	131	9	can	can	AUX
ejpam-4835	131	10	be	be	AUX
ejpam-4835	131	11	established	establish	VERB
ejpam-4835	131	12	:	:	PUNCT
ejpam-4835	131	13	∥(aν	∥(aν	ADJ
ejpam-4835	131	14	k)s	k)s	PROPN
ejpam-4835	132	1	α	α	X
ejpam-4835	132	2	k	k	X
ejpam-4835	132	3	∥∞	∥∞	ADJ
ejpam-4835	132	4	≤	≤	NUM
ejpam-4835	132	5	c	c	NOUN
ejpam-4835	132	6	1√	1√	NUM
ejpam-4835	132	7	α	α	NOUN
ejpam-4835	132	8	h−2	h−2	PROPN
ejpam-4835	132	9	k	k	PROPN
ejpam-4835	132	10	.	.	PUNCT
ejpam-4835	133	1	(	(	PUNCT
ejpam-4835	133	2	3.4	3.4	NUM
ejpam-4835	133	3	)	)	PUNCT
ejpam-4835	133	4	by	by	ADP
ejpam-4835	133	5	passing	pass	VERB
ejpam-4835	133	6	to	to	ADP
ejpam-4835	133	7	the	the	DET
ejpam-4835	133	8	norm	norm	NOUN
ejpam-4835	133	9	in	in	ADP
ejpam-4835	133	10	(	(	PUNCT
ejpam-4835	133	11	2.13	2.13	NUM
ejpam-4835	133	12	)	)	PUNCT
ejpam-4835	133	13	,	,	PUNCT
ejpam-4835	133	14	and	and	CCONJ
ejpam-4835	133	15	taking	take	VERB
ejpam-4835	133	16	into	into	ADP
ejpam-4835	133	17	account	account	NOUN
ejpam-4835	133	18	(	(	PUNCT
ejpam-4835	133	19	3.1	3.1	NUM
ejpam-4835	133	20	)	)	PUNCT
ejpam-4835	133	21	and	and	CCONJ
ejpam-4835	133	22	(	(	PUNCT
ejpam-4835	133	23	3.4	3.4	NUM
ejpam-4835	133	24	)	)	PUNCT
ejpam-4835	133	25	,	,	PUNCT
ejpam-4835	133	26	we	we	PRON
ejpam-4835	133	27	have	have	VERB
ejpam-4835	133	28	to	to	PART
ejpam-4835	133	29	prove	prove	VERB
ejpam-4835	133	30	the	the	DET
ejpam-4835	133	31	the	the	DET
ejpam-4835	133	32	following	follow	VERB
ejpam-4835	133	33	stability	stability	NOUN
ejpam-4835	133	34	limit	limit	NOUN
ejpam-4835	133	35	:	:	PUNCT
ejpam-4835	133	36	∃cs	∃cs	ADJ
ejpam-4835	133	37	:	:	PUNCT
ejpam-4835	133	38	∥sα	∥sα	ADJ
ejpam-4835	133	39	k	k	X
ejpam-4835	133	40	∥∞	∥∞	ADJ
ejpam-4835	133	41	≤	≤	NUM
ejpam-4835	133	42	cs	cs	PROPN
ejpam-4835	133	43	,	,	PUNCT
ejpam-4835	133	44	for	for	ADP
ejpam-4835	133	45	all	all	DET
ejpam-4835	133	46	k	k	PROPN
ejpam-4835	133	47	and	and	CCONJ
ejpam-4835	133	48	α	α	NOUN
ejpam-4835	133	49	.	.	PUNCT
ejpam-4835	134	1	(	(	PUNCT
ejpam-4835	134	2	3.5	3.5	NUM
ejpam-4835	134	3	)	)	PUNCT
ejpam-4835	134	4	the	the	DET
ejpam-4835	134	5	convergence	convergence	NOUN
ejpam-4835	134	6	analysis	analysis	NOUN
ejpam-4835	134	7	is	be	AUX
ejpam-4835	134	8	performed	perform	VERB
ejpam-4835	134	9	based	base	VERB
ejpam-4835	134	10	on	on	ADP
ejpam-4835	134	11	the	the	DET
ejpam-4835	134	12	split	split	NOUN
ejpam-4835	134	13	of	of	ADP
ejpam-4835	134	14	the	the	DET
ejpam-4835	134	15	following	follow	VERB
ejpam-4835	134	16	two	two	NUM
ejpam-4835	134	17	lattices	lattice	NOUN
ejpam-4835	134	18	:	:	PUNCT
ejpam-4835	134	19	the	the	DET
ejpam-4835	134	20	iterate	iterate	NOUN
ejpam-4835	134	21	matrix	matrix	NOUN
ejpam-4835	134	22	with	with	ADP
ejpam-4835	134	23	α2	α2	PROPN
ejpam-4835	134	24	=	=	SYM
ejpam-4835	135	1	0	0	X
ejpam-4835	135	2	.	.	X
ejpam-4835	135	3	∥tgk(α1	∥tgk(α1	NOUN
ejpam-4835	135	4	,	,	PUNCT
ejpam-4835	135	5	0)∥∞	0)∥∞	X
ejpam-4835	135	6	=	=	SYM
ejpam-4835	135	7	∥∥∥((aν	∥∥∥((aν	PROPN
ejpam-4835	135	8	k	k	NOUN
ejpam-4835	135	9	)	)	PUNCT
ejpam-4835	135	10	−1	−1	NOUN
ejpam-4835	135	11	−	−	PROPN
ejpam-4835	135	12	pk	pk	NOUN
ejpam-4835	135	13	(	(	PUNCT
ejpam-4835	135	14	aν	aν	NOUN
ejpam-4835	135	15	k−1	k−1	PROPN
ejpam-4835	135	16	)	)	PUNCT
ejpam-4835	135	17	−1rk	−1rk	NOUN
ejpam-4835	135	18	)	)	PUNCT
ejpam-4835	135	19	(	(	PUNCT
ejpam-4835	135	20	aν	aν	NOUN
ejpam-4835	135	21	k)s	k)s	NOUN
ejpam-4835	135	22	α1	α1	PROPN
ejpam-4835	135	23	k	k	PROPN
ejpam-4835	135	24	∥∥∥	∥∥∥	PROPN
ejpam-4835	135	25	∞	∞	NUM
ejpam-4835	135	26	≤	≤	NUM
ejpam-4835	136	1	∥∥∥((aν	∥∥∥((aν	PROPN
ejpam-4835	136	2	k	k	NOUN
ejpam-4835	136	3	)	)	PUNCT
ejpam-4835	136	4	−1	−1	NOUN
ejpam-4835	136	5	−	−	PROPN
ejpam-4835	136	6	pk	pk	NOUN
ejpam-4835	136	7	(	(	PUNCT
ejpam-4835	136	8	aν	aν	NOUN
ejpam-4835	136	9	k−1	k−1	PROPN
ejpam-4835	136	10	)	)	PUNCT
ejpam-4835	136	11	−1rk	−1rk	NOUN
ejpam-4835	136	12	)	)	PUNCT
ejpam-4835	136	13	∥∥∥	∥∥∥	PROPN
ejpam-4835	136	14	∞	∞	NUM
ejpam-4835	136	15	∥∥(aν	∥∥(aν	PROPN
ejpam-4835	137	1	k)s	k)s	PROPN
ejpam-4835	137	2	α1	α1	PROPN
ejpam-4835	137	3	k	k	X
ejpam-4835	137	4	∥∥	∥∥	X
ejpam-4835	137	5	∞	∞	NUM
ejpam-4835	137	6	.	.	PUNCT
ejpam-4835	138	1	typically	typically	ADV
ejpam-4835	138	2	,	,	PUNCT
ejpam-4835	138	3	a	a	DET
ejpam-4835	138	4	hierarchy	hierarchy	NOUN
ejpam-4835	138	5	of	of	ADP
ejpam-4835	138	6	more	more	ADJ
ejpam-4835	138	7	than	than	ADP
ejpam-4835	138	8	two	two	NUM
ejpam-4835	138	9	grids	grid	NOUN
ejpam-4835	138	10	is	be	AUX
ejpam-4835	138	11	selected	select	VERB
ejpam-4835	138	12	.	.	PUNCT
ejpam-4835	139	1	in	in	ADP
ejpam-4835	139	2	this	this	DET
ejpam-4835	139	3	scenario	scenario	NOUN
ejpam-4835	139	4	,	,	PUNCT
ejpam-4835	139	5	iterative	iterative	NOUN
ejpam-4835	139	6	matrices	matrix	NOUN
ejpam-4835	139	7	(	(	PUNCT
ejpam-4835	139	8	2.16	2.16	NUM
ejpam-4835	139	9	)	)	PUNCT
ejpam-4835	139	10	can	can	AUX
ejpam-4835	139	11	be	be	AUX
ejpam-4835	139	12	defined	define	VERB
ejpam-4835	139	13	using	use	VERB
ejpam-4835	139	14	recursion	recursion	NOUN
ejpam-4835	139	15	of	of	ADP
ejpam-4835	139	16	iterative	iterative	ADJ
ejpam-4835	139	17	matrices	matrix	NOUN
ejpam-4835	139	18	(	(	PUNCT
ejpam-4835	139	19	2.15	2.15	NUM
ejpam-4835	139	20	)	)	PUNCT
ejpam-4835	139	21	for	for	ADP
ejpam-4835	139	22	all	all	DET
ejpam-4835	139	23	levels	level	NOUN
ejpam-4835	139	24	k.	k.	INTJ
ejpam-4835	140	1	if	if	SCONJ
ejpam-4835	140	2	we	we	PRON
ejpam-4835	140	3	h.	h.	PROPN
ejpam-4835	140	4	ahmad	ahmad	PROPN
ejpam-4835	140	5	et	et	PROPN
ejpam-4835	140	6	al	al	PROPN
ejpam-4835	140	7	.	.	PUNCT
ejpam-4835	140	8	/	/	SYM
ejpam-4835	140	9	eur	eur	PROPN
ejpam-4835	140	10	.	.	PUNCT
ejpam-4835	141	1	j.	j.	PROPN
ejpam-4835	141	2	pure	pure	PROPN
ejpam-4835	141	3	appl	appl	PROPN
ejpam-4835	141	4	.	.	PROPN
ejpam-4835	141	5	math	math	PROPN
ejpam-4835	141	6	,	,	PUNCT
ejpam-4835	141	7	16	16	NUM
ejpam-4835	141	8	(	(	PUNCT
ejpam-4835	141	9	3	3	NUM
ejpam-4835	141	10	)	)	PUNCT
ejpam-4835	141	11	(	(	PUNCT
ejpam-4835	141	12	2023	2023	NUM
ejpam-4835	141	13	)	)	PUNCT
ejpam-4835	141	14	,	,	PUNCT
ejpam-4835	141	15	1956	1956	NUM
ejpam-4835	141	16	-	-	SYM
ejpam-4835	141	17	1969	1969	NUM
ejpam-4835	141	18	1963	1963	NUM
ejpam-4835	141	19	assume	assume	VERB
ejpam-4835	141	20	that	that	SCONJ
ejpam-4835	141	21	(	(	PUNCT
ejpam-4835	141	22	3.5	3.5	NUM
ejpam-4835	141	23	)	)	PUNCT
ejpam-4835	141	24	holds	hold	VERB
ejpam-4835	141	25	,	,	PUNCT
ejpam-4835	141	26	then	then	ADV
ejpam-4835	141	27	l∞-convergence	l∞-convergence	NOUN
ejpam-4835	141	28	results	result	NOUN
ejpam-4835	141	29	can	can	AUX
ejpam-4835	141	30	be	be	AUX
ejpam-4835	141	31	readily	readily	ADV
ejpam-4835	141	32	deduced	deduce	VERB
ejpam-4835	141	33	from	from	ADP
ejpam-4835	141	34	the	the	DET
ejpam-4835	141	35	previous	previous	ADJ
ejpam-4835	141	36	findings	finding	NOUN
ejpam-4835	141	37	.	.	PUNCT
ejpam-4835	142	1	theorem	theorem	VERB
ejpam-4835	142	2	8	8	NUM
ejpam-4835	142	3	:	:	PUNCT
ejpam-4835	142	4	(	(	PUNCT
ejpam-4835	142	5	[	[	X
ejpam-4835	142	6	26	26	NUM
ejpam-4835	142	7	]	]	PUNCT
ejpam-4835	142	8	)	)	PUNCT
ejpam-4835	142	9	consider	consider	VERB
ejpam-4835	142	10	a	a	DET
ejpam-4835	142	11	multigrid	multigrid	NOUN
ejpam-4835	142	12	method	method	NOUN
ejpam-4835	142	13	for	for	ADP
ejpam-4835	142	14	a	a	DET
ejpam-4835	142	15	given	give	VERB
ejpam-4835	142	16	iterative	iterative	NOUN
ejpam-4835	142	17	matrix	matrix	NOUN
ejpam-4835	142	18	(	(	PUNCT
ejpam-4835	142	19	2.16	2.16	NUM
ejpam-4835	142	20	)	)	PUNCT
ejpam-4835	142	21	.	.	PUNCT
ejpam-4835	143	1	then	then	ADV
ejpam-4835	143	2	under	under	ADP
ejpam-4835	143	3	the	the	DET
ejpam-4835	143	4	previous	previous	ADJ
ejpam-4835	143	5	assumption	assumption	NOUN
ejpam-4835	143	6	,	,	PUNCT
ejpam-4835	143	7	for	for	ADP
ejpam-4835	143	8	the	the	DET
ejpam-4835	143	9	parameter	parameter	NOUN
ejpam-4835	143	10	value	value	NOUN
ejpam-4835	143	11	α2	α2	PROPN
ejpam-4835	143	12	=	=	SYM
ejpam-4835	143	13	0	0	NUM
ejpam-4835	143	14	,	,	PUNCT
ejpam-4835	143	15	α1	α1	PROPN
ejpam-4835	143	16	=	=	SYM
ejpam-4835	143	17	α	α	PROPN
ejpam-4835	143	18	>	>	X
ejpam-4835	143	19	0	0	PROPN
ejpam-4835	143	20	,	,	PUNCT
ejpam-4835	143	21	τ	τ	X
ejpam-4835	143	22	≥	≥	NOUN
ejpam-4835	143	23	2	2	NUM
ejpam-4835	143	24	.	.	PUNCT
ejpam-4835	143	25	for	for	ADP
ejpam-4835	143	26	each	each	DET
ejpam-4835	143	27	ζ	ζ	PROPN
ejpam-4835	143	28	∈	∈	NOUN
ejpam-4835	143	29	(	(	PUNCT
ejpam-4835	143	30	0	0	NUM
ejpam-4835	143	31	,	,	PUNCT
ejpam-4835	143	32	1	1	X
ejpam-4835	143	33	)	)	PUNCT
ejpam-4835	143	34	there	there	PRON
ejpam-4835	143	35	is	be	VERB
ejpam-4835	143	36	α	α	NOUN
ejpam-4835	143	37	,	,	PUNCT
ejpam-4835	143	38	α∗such	α∗such	ADP
ejpam-4835	143	39	that	that	PRON
ejpam-4835	143	40	for	for	ADP
ejpam-4835	143	41	all	all	PRON
ejpam-4835	143	42	α	α	PRON
ejpam-4835	143	43	≥	≥	NOUN
ejpam-4835	143	44	α∗	α∗	VERB
ejpam-4835	143	45	∥mgk∥∞	∥mgk∥∞	PROPN
ejpam-4835	143	46	≤	≤	NUM
ejpam-4835	143	47	ζ	ζ	NOUN
ejpam-4835	143	48	,	,	PUNCT
ejpam-4835	143	49	k	k	NOUN
ejpam-4835	143	50	=	=	SYM
ejpam-4835	143	51	0	0	NUM
ejpam-4835	143	52	,	,	PUNCT
ejpam-4835	143	53	1	1	NUM
ejpam-4835	143	54	,	,	PUNCT
ejpam-4835	143	55	...	...	PUNCT
ejpam-4835	143	56	(	(	PUNCT
ejpam-4835	143	57	3.7	3.7	NUM
ejpam-4835	143	58	)	)	PUNCT
ejpam-4835	143	59	hold	hold	VERB
ejpam-4835	143	60	.	.	PUNCT
ejpam-4835	144	1	proof	proof	NOUN
ejpam-4835	144	2	.	.	PUNCT
ejpam-4835	145	1	if	if	SCONJ
ejpam-4835	145	2	the	the	DET
ejpam-4835	145	3	approximation	approximation	NOUN
ejpam-4835	145	4	and	and	CCONJ
ejpam-4835	145	5	smoothness	smoothness	ADJ
ejpam-4835	145	6	properties	property	NOUN
ejpam-4835	145	7	are	be	AUX
ejpam-4835	145	8	combined	combine	VERB
ejpam-4835	145	9	with	with	ADP
ejpam-4835	145	10	(	(	PUNCT
ejpam-4835	145	11	3.5	3.5	NUM
ejpam-4835	145	12	)	)	PUNCT
ejpam-4835	145	13	,	,	PUNCT
ejpam-4835	145	14	then	then	ADV
ejpam-4835	145	15	we	we	PRON
ejpam-4835	145	16	can	can	AUX
ejpam-4835	145	17	apply	apply	VERB
ejpam-4835	145	18	the	the	DET
ejpam-4835	145	19	same	same	ADJ
ejpam-4835	145	20	parameters	parameter	NOUN
ejpam-4835	145	21	as	as	ADP
ejpam-4835	145	22	in	in	ADP
ejpam-4835	145	23	[	[	X
ejpam-4835	145	24	26	26	NUM
ejpam-4835	145	25	]	]	PUNCT
ejpam-4835	145	26	.	.	PUNCT
ejpam-4835	146	1	the	the	DET
ejpam-4835	146	2	following	follow	VERB
ejpam-4835	146	3	theorem	theorem	NOUN
ejpam-4835	146	4	represents	represent	VERB
ejpam-4835	146	5	the	the	DET
ejpam-4835	146	6	main	main	ADJ
ejpam-4835	146	7	result	result	NOUN
ejpam-4835	146	8	of	of	ADP
ejpam-4835	146	9	our	our	PRON
ejpam-4835	146	10	work	work	NOUN
ejpam-4835	146	11	.	.	PUNCT
ejpam-4835	147	1	theorem	theorem	VERB
ejpam-4835	147	2	9	9	NUM
ejpam-4835	147	3	:	:	PUNCT
ejpam-4835	147	4	under	under	ADP
ejpam-4835	147	5	the	the	DET
ejpam-4835	147	6	previous	previous	ADJ
ejpam-4835	147	7	assumptions	assumption	NOUN
ejpam-4835	147	8	and	and	CCONJ
ejpam-4835	147	9	notations	notation	NOUN
ejpam-4835	147	10	the	the	DET
ejpam-4835	147	11	iterated	iterated	ADJ
ejpam-4835	147	12	uνk	uνk	PROPN
ejpam-4835	147	13	,	,	PUNCT
ejpam-4835	147	14	ν	ν	X
ejpam-4835	147	15	≥	≥	X
ejpam-4835	147	16	0	0	NUM
ejpam-4835	147	17	for	for	ADP
ejpam-4835	147	18	two	two	NUM
ejpam-4835	147	19	meshes	mesh	NOUN
ejpam-4835	147	20	k	k	PROPN
ejpam-4835	147	21	and	and	CCONJ
ejpam-4835	147	22	k	k	PROPN
ejpam-4835	147	23	−	−	PROPN
ejpam-4835	147	24	1	1	NUM
ejpam-4835	147	25	satisfy	satisfy	NOUN
ejpam-4835	147	26	:	:	PUNCT
ejpam-4835	147	27	∥∥uν+1	∥∥uν+1	PROPN
ejpam-4835	147	28	k	k	PROPN
ejpam-4835	148	1	−	−	PROPN
ejpam-4835	148	2	u∗k	u∗k	ADJ
ejpam-4835	148	3	∥∥	∥∥	X
ejpam-4835	148	4	∞	∞	NUM
ejpam-4835	148	5	≤	≤	NUM
ejpam-4835	148	6	(	(	PUNCT
ejpam-4835	148	7	c√	c√	NUM
ejpam-4835	148	8	α	α	PRON
ejpam-4835	148	9	|loghk|	|loghk|	NUM
ejpam-4835	148	10	2	2	NUM
ejpam-4835	148	11	)	)	PUNCT
ejpam-4835	149	1	∥uνk	∥uνk	VERB
ejpam-4835	149	2	−	−	NOUN
ejpam-4835	149	3	u∗k∥∞	u∗k∥∞	PROPN
ejpam-4835	149	4	.	.	PUNCT
ejpam-4835	150	1	(	(	PUNCT
ejpam-4835	150	2	3.6	3.6	NUM
ejpam-4835	150	3	)	)	PUNCT
ejpam-4835	150	4	proof	proof	NOUN
ejpam-4835	150	5	.	.	PUNCT
ejpam-4835	151	1	we	we	PRON
ejpam-4835	151	2	have∥∥uν+1	have∥∥uν+1	VERB
ejpam-4835	151	3	k	k	NOUN
ejpam-4835	152	1	−	−	PROPN
ejpam-4835	153	1	u∗k	u∗k	ADJ
ejpam-4835	153	2	∥∥	∥∥	X
ejpam-4835	153	3	∞	∞	NUM
ejpam-4835	153	4	=	=	SYM
ejpam-4835	153	5	∥∥∥((ik	∥∥∥((ik	PROPN
ejpam-4835	153	6	−	−	PROPN
ejpam-4835	153	7	pk	pk	NOUN
ejpam-4835	153	8	(	(	PUNCT
ejpam-4835	153	9	ik	ik	PROPN
ejpam-4835	153	10	−mgk−1	−mgk−1	PROPN
ejpam-4835	153	11	)	)	PUNCT
ejpam-4835	154	1	(	(	PUNCT
ejpam-4835	154	2	aν	aν	NOUN
ejpam-4835	154	3	k−1	k−1	PROPN
ejpam-4835	154	4	)	)	PUNCT
ejpam-4835	154	5	−1rk	−1rk	NOUN
ejpam-4835	154	6	)	)	PUNCT
ejpam-4835	155	1	(	(	PUNCT
ejpam-4835	155	2	aν	aν	NOUN
ejpam-4835	155	3	k)s	k)s	X
ejpam-4835	155	4	α1	α1	PROPN
ejpam-4835	155	5	k	k	PROPN
ejpam-4835	155	6	)	)	PUNCT
ejpam-4835	155	7	(	(	PUNCT
ejpam-4835	155	8	uνk	uνk	ADJ
ejpam-4835	155	9	−	−	NOUN
ejpam-4835	155	10	u∗k	u∗k	ADJ
ejpam-4835	155	11	)	)	PUNCT
ejpam-4835	155	12	∥∥∥	∥∥∥	NUM
ejpam-4835	155	13	∞	∞	NUM
ejpam-4835	155	14	≤	≤	NUM
ejpam-4835	156	1	∥∥∥(ik	∥∥∥(ik	PROPN
ejpam-4835	156	2	−	−	PROPN
ejpam-4835	156	3	pk	pk	NOUN
ejpam-4835	156	4	(	(	PUNCT
ejpam-4835	156	5	ik	ik	PROPN
ejpam-4835	156	6	−mgk−1	−mgk−1	PROPN
ejpam-4835	156	7	)	)	PUNCT
ejpam-4835	156	8	(	(	PUNCT
ejpam-4835	156	9	aν	aν	NOUN
ejpam-4835	156	10	k−1	k−1	PROPN
ejpam-4835	156	11	)	)	PUNCT
ejpam-4835	156	12	−1rk	−1rk	NOUN
ejpam-4835	156	13	)	)	PUNCT
ejpam-4835	156	14	∥∥∥	∥∥∥	PROPN
ejpam-4835	156	15	∞	∞	PROPN
ejpam-4835	156	16	∥∥aν	∥∥aν	PROPN
ejpam-4835	157	1	ks	ks	PROPN
ejpam-4835	157	2	α1	α1	PROPN
ejpam-4835	157	3	k	k	X
ejpam-4835	157	4	∥∥	∥∥	X
ejpam-4835	157	5	∞	∞	NUM
ejpam-4835	158	1	∥uνk	∥uνk	VERB
ejpam-4835	158	2	−	−	NOUN
ejpam-4835	158	3	u∗k∥∞	u∗k∥∞	PROPN
ejpam-4835	158	4	≤	≤	PROPN
ejpam-4835	158	5	(	(	PUNCT
ejpam-4835	158	6	c2√	c2√	NOUN
ejpam-4835	158	7	α	α	NOUN
ejpam-4835	158	8	h−2	h−2	PROPN
ejpam-4835	158	9	k	k	PROPN
ejpam-4835	158	10	)	)	PUNCT
ejpam-4835	158	11	(	(	PUNCT
ejpam-4835	158	12	c1h	c1h	PROPN
ejpam-4835	158	13	2	2	NUM
ejpam-4835	158	14	k	k	PROPN
ejpam-4835	158	15	|log	|log	X
ejpam-4835	158	16	hk|	hk|	PROPN
ejpam-4835	158	17	2	2	NUM
ejpam-4835	158	18	)	)	PUNCT
ejpam-4835	158	19	∥(uνk	∥(uνk	ADP
ejpam-4835	158	20	−	−	PROPN
ejpam-4835	158	21	u∗k)∥∞	u∗k)∥∞	PROPN
ejpam-4835	158	22	≤	≤	PROPN
ejpam-4835	159	1	(	(	PUNCT
ejpam-4835	159	2	c1c2√	c1c2√	NOUN
ejpam-4835	159	3	α	α	PROPN
ejpam-4835	159	4	)	)	PUNCT
ejpam-4835	159	5	|log	|log	PROPN
ejpam-4835	160	1	hk|2	hk|2	PROPN
ejpam-4835	160	2	∥uνk	∥uνk	VERB
ejpam-4835	160	3	−	−	NOUN
ejpam-4835	160	4	u∗k∥∞	u∗k∥∞	PROPN
ejpam-4835	160	5	4	4	NUM
ejpam-4835	160	6	.	.	PUNCT
ejpam-4835	161	1	numerical	numerical	PROPN
ejpam-4835	161	2	simulation	simulation	PROPN
ejpam-4835	161	3	in	in	ADP
ejpam-4835	161	4	this	this	DET
ejpam-4835	161	5	section	section	NOUN
ejpam-4835	161	6	,	,	PUNCT
ejpam-4835	161	7	we	we	PRON
ejpam-4835	161	8	present	present	VERB
ejpam-4835	161	9	a	a	DET
ejpam-4835	161	10	numerical	numerical	ADJ
ejpam-4835	161	11	example	example	NOUN
ejpam-4835	161	12	of	of	ADP
ejpam-4835	161	13	a	a	DET
ejpam-4835	161	14	nonlinear	nonlinear	ADJ
ejpam-4835	161	15	variational	variational	ADJ
ejpam-4835	161	16	inequality	inequality	NOUN
ejpam-4835	161	17	.	.	PUNCT
ejpam-4835	162	1	to	to	PART
ejpam-4835	162	2	apply	apply	VERB
ejpam-4835	162	3	the	the	DET
ejpam-4835	162	4	proposed	propose	VERB
ejpam-4835	162	5	method	method	NOUN
ejpam-4835	162	6	to	to	ADP
ejpam-4835	162	7	the	the	DET
ejpam-4835	162	8	example	example	NOUN
ejpam-4835	162	9	,	,	PUNCT
ejpam-4835	162	10	we	we	PRON
ejpam-4835	162	11	assume	assume	VERB
ejpam-4835	162	12	that	that	SCONJ
ejpam-4835	162	13	the	the	DET
ejpam-4835	162	14	data	datum	NOUN
ejpam-4835	162	15	of	of	ADP
ejpam-4835	162	16	the	the	DET
ejpam-4835	162	17	problem	problem	NOUN
ejpam-4835	162	18	is	be	AUX
ejpam-4835	162	19	sufficiently	sufficiently	ADV
ejpam-4835	162	20	smooth	smooth	ADJ
ejpam-4835	162	21	.	.	PUNCT
ejpam-4835	163	1	we	we	PRON
ejpam-4835	163	2	then	then	ADV
ejpam-4835	163	3	employ	employ	VERB
ejpam-4835	163	4	bellman	bellman	NOUN
ejpam-4835	163	5	’s	’s	PART
ejpam-4835	163	6	principle	principle	NOUN
ejpam-4835	163	7	of	of	ADP
ejpam-4835	163	8	dynamic	dynamic	ADJ
ejpam-4835	163	9	programming	programming	NOUN
ejpam-4835	163	10	to	to	PART
ejpam-4835	163	11	tackle	tackle	VERB
ejpam-4835	163	12	the	the	DET
ejpam-4835	163	13	problem	problem	NOUN
ejpam-4835	163	14	and	and	CCONJ
ejpam-4835	163	15	proceed	proceed	VERB
ejpam-4835	163	16	with	with	ADP
ejpam-4835	163	17	solving	solve	VERB
ejpam-4835	163	18	(	(	PUNCT
ejpam-4835	163	19	2.3	2.3	NUM
ejpam-4835	163	20	)	)	PUNCT
ejpam-4835	163	21	,	,	PUNCT
ejpam-4835	163	22	as	as	SCONJ
ejpam-4835	163	23	discussed	discuss	VERB
ejpam-4835	163	24	previously	previously	ADV
ejpam-4835	163	25	,	,	PUNCT
ejpam-4835	163	26	using	use	VERB
ejpam-4835	163	27	the	the	DET
ejpam-4835	163	28	following	follow	VERB
ejpam-4835	163	29	specified	specify	VERB
ejpam-4835	163	30	data:	data:	PROPN
ejpam-4835	163	31	au	au	X
ejpam-4835	163	32	≤	≤	PROPN
ejpam-4835	163	33	f(u	f(u	PROPN
ejpam-4835	163	34	)	)	PUNCT
ejpam-4835	163	35	,	,	PUNCT
ejpam-4835	163	36	in	in	ADP
ejpam-4835	163	37	ω	ω	NUM
ejpam-4835	163	38	=	=	SYM
ejpam-4835	163	39	{	{	PUNCT
ejpam-4835	163	40	(	(	PUNCT
ejpam-4835	163	41	x	x	NOUN
ejpam-4835	163	42	,	,	PUNCT
ejpam-4835	163	43	y	y	NOUN
ejpam-4835	163	44	)	)	PUNCT
ejpam-4835	164	1	|	|	ADV
ejpam-4835	164	2	x2	x2	INTJ
ejpam-4835	165	1	+	+	CCONJ
ejpam-4835	165	2	y2	y2	ADJ
ejpam-4835	165	3	≤	≤	ADJ
ejpam-4835	165	4	1	1	NUM
ejpam-4835	165	5	}	}	PUNCT
ejpam-4835	165	6	,	,	PUNCT
ejpam-4835	165	7	⟨au−	⟨au−	NOUN
ejpam-4835	165	8	f(u	f(u	PROPN
ejpam-4835	165	9	)	)	PUNCT
ejpam-4835	165	10	,	,	PUNCT
ejpam-4835	165	11	u−	u−	PROPN
ejpam-4835	165	12	ψ⟩	ψ⟩	X
ejpam-4835	165	13	=	=	NOUN
ejpam-4835	165	14	0	0	NUM
ejpam-4835	165	15	,	,	PUNCT
ejpam-4835	165	16	u	u	NOUN
ejpam-4835	165	17	≤	≤	X
ejpam-4835	165	18	ψ	ψ	X
ejpam-4835	165	19	,	,	PUNCT
ejpam-4835	165	20	u	u	NOUN
ejpam-4835	165	21	=	=	PROPN
ejpam-4835	165	22	0	0	NUM
ejpam-4835	165	23	,	,	PUNCT
ejpam-4835	165	24	in	in	ADP
ejpam-4835	165	25	∂ω	∂ω	PROPN
ejpam-4835	165	26	,	,	PUNCT
ejpam-4835	165	27	(	(	PUNCT
ejpam-4835	165	28	4.1	4.1	NUM
ejpam-4835	165	29	)	)	PUNCT
ejpam-4835	166	1	h.	h.	PROPN
ejpam-4835	166	2	ahmad	ahmad	PROPN
ejpam-4835	166	3	et	et	PROPN
ejpam-4835	166	4	al	al	PROPN
ejpam-4835	166	5	.	.	PUNCT
ejpam-4835	166	6	/	/	SYM
ejpam-4835	166	7	eur	eur	PROPN
ejpam-4835	166	8	.	.	PUNCT
ejpam-4835	167	1	j.	j.	PROPN
ejpam-4835	167	2	pure	pure	PROPN
ejpam-4835	167	3	appl	appl	PROPN
ejpam-4835	167	4	.	.	PROPN
ejpam-4835	167	5	math	math	PROPN
ejpam-4835	167	6	,	,	PUNCT
ejpam-4835	167	7	16	16	NUM
ejpam-4835	167	8	(	(	PUNCT
ejpam-4835	167	9	3	3	NUM
ejpam-4835	167	10	)	)	PUNCT
ejpam-4835	167	11	(	(	PUNCT
ejpam-4835	167	12	2023	2023	NUM
ejpam-4835	167	13	)	)	PUNCT
ejpam-4835	167	14	,	,	PUNCT
ejpam-4835	167	15	1956	1956	NUM
ejpam-4835	167	16	-	-	SYM
ejpam-4835	167	17	1969	1969	NUM
ejpam-4835	167	18	1964	1964	NUM
ejpam-4835	167	19	where	where	SCONJ
ejpam-4835	167	20	au	au	ADV
ejpam-4835	167	21	=	=	SYM
ejpam-4835	167	22	−∆u	−∆u	NOUN
ejpam-4835	167	23	,	,	PUNCT
ejpam-4835	167	24	f(u	f(u	PROPN
ejpam-4835	167	25	)	)	PUNCT
ejpam-4835	167	26	=	=	SYM
ejpam-4835	168	1	cosu	cosu	NOUN
ejpam-4835	168	2	,	,	PUNCT
ejpam-4835	168	3	ψ	ψ	X
ejpam-4835	168	4	=	=	NOUN
ejpam-4835	168	5	0	0	X
ejpam-4835	168	6	.	.	PUNCT
ejpam-4835	169	1	we	we	PRON
ejpam-4835	169	2	are	be	AUX
ejpam-4835	169	3	confine	confine	VERB
ejpam-4835	169	4	ourselves	ourselves	PRON
ejpam-4835	169	5	to	to	ADP
ejpam-4835	169	6	the	the	DET
ejpam-4835	169	7	fem	fem	NOUN
ejpam-4835	169	8	discretization	discretization	NOUN
ejpam-4835	169	9	with	with	ADP
ejpam-4835	169	10	a	a	DET
ejpam-4835	169	11	uniform	uniform	ADJ
ejpam-4835	169	12	triangulation	triangulation	NOUN
ejpam-4835	169	13	and	and	CCONJ
ejpam-4835	169	14	p1	p1	PROPN
ejpam-4835	169	15	nested	nest	VERB
ejpam-4835	169	16	finite	finite	PROPN
ejpam-4835	169	17	element	element	NOUN
ejpam-4835	169	18	function	function	NOUN
ejpam-4835	169	19	spaces	space	VERB
ejpam-4835	169	20	.	.	PUNCT
ejpam-4835	170	1	for	for	ADP
ejpam-4835	170	2	the	the	DET
ejpam-4835	170	3	discretization	discretization	NOUN
ejpam-4835	170	4	of	of	ADP
ejpam-4835	170	5	the	the	DET
ejpam-4835	170	6	domain	domain	NOUN
ejpam-4835	170	7	,	,	PUNCT
ejpam-4835	170	8	we	we	PRON
ejpam-4835	170	9	have	have	AUX
ejpam-4835	170	10	used	use	VERB
ejpam-4835	170	11	the	the	DET
ejpam-4835	170	12	pde	pde	NOUN
ejpam-4835	170	13	toolbox	toolbox	NOUN
ejpam-4835	170	14	in	in	ADP
ejpam-4835	170	15	matlab	matlab	PROPN
ejpam-4835	170	16	(	(	PUNCT
ejpam-4835	170	17	r2017b	r2017b	PROPN
ejpam-4835	170	18	)	)	PUNCT
ejpam-4835	170	19	to	to	PART
ejpam-4835	170	20	generate	generate	VERB
ejpam-4835	170	21	the	the	DET
ejpam-4835	170	22	mesh	mesh	NOUN
ejpam-4835	170	23	and	and	CCONJ
ejpam-4835	170	24	then	then	ADV
ejpam-4835	170	25	the	the	DET
ejpam-4835	170	26	multigrid	multigrid	NOUN
ejpam-4835	170	27	fem	fem	NOUN
ejpam-4835	170	28	can	can	AUX
ejpam-4835	170	29	be	be	AUX
ejpam-4835	170	30	used	use	VERB
ejpam-4835	170	31	to	to	PART
ejpam-4835	170	32	efficiently	efficiently	ADV
ejpam-4835	170	33	solve	solve	VERB
ejpam-4835	170	34	4.1	4.1	NUM
ejpam-4835	170	35	as	as	SCONJ
ejpam-4835	170	36	mentioned	mention	VERB
ejpam-4835	170	37	above	above	ADV
ejpam-4835	170	38	.	.	PUNCT
ejpam-4835	171	1	figure	figure	VERB
ejpam-4835	171	2	1	1	NUM
ejpam-4835	171	3	:	:	PUNCT
ejpam-4835	171	4	domain	domain	NOUN
ejpam-4835	171	5	of	of	ADP
ejpam-4835	171	6	our	our	PRON
ejpam-4835	171	7	problem	problem	NOUN
ejpam-4835	171	8	with	with	ADP
ejpam-4835	171	9	2048	2048	NUM
ejpam-4835	171	10	triangle	triangle	NOUN
ejpam-4835	171	11	and	and	CCONJ
ejpam-4835	171	12	1089	1089	NUM
ejpam-4835	171	13	nodes	node	NOUN
ejpam-4835	171	14	.	.	PUNCT
ejpam-4835	172	1	this	this	DET
ejpam-4835	172	2	numerical	numerical	ADJ
ejpam-4835	172	3	example	example	NOUN
ejpam-4835	172	4	is	be	AUX
ejpam-4835	172	5	conducted	conduct	VERB
ejpam-4835	172	6	to	to	PART
ejpam-4835	172	7	demonstrate	demonstrate	VERB
ejpam-4835	172	8	the	the	DET
ejpam-4835	172	9	high	high	ADJ
ejpam-4835	172	10	efficiency	efficiency	NOUN
ejpam-4835	172	11	of	of	ADP
ejpam-4835	172	12	the	the	DET
ejpam-4835	172	13	multigrid	multigrid	NOUN
ejpam-4835	172	14	method	method	NOUN
ejpam-4835	172	15	.	.	PUNCT
ejpam-4835	173	1	in	in	ADP
ejpam-4835	173	2	this	this	DET
ejpam-4835	173	3	study	study	NOUN
ejpam-4835	173	4	,	,	PUNCT
ejpam-4835	173	5	we	we	PRON
ejpam-4835	173	6	utilize	utilize	VERB
ejpam-4835	173	7	the	the	DET
ejpam-4835	173	8	gauss	gauss	ADJ
ejpam-4835	173	9	-	-	PUNCT
ejpam-4835	173	10	seidel	seidel	PROPN
ejpam-4835	173	11	method	method	NOUN
ejpam-4835	173	12	for	for	ADP
ejpam-4835	173	13	pre	pre	ADJ
ejpam-4835	173	14	/	/	SYM
ejpam-4835	173	15	post	post	ADJ
ejpam-4835	173	16	smoothing	smoothing	NOUN
ejpam-4835	173	17	within	within	ADP
ejpam-4835	173	18	the	the	DET
ejpam-4835	173	19	multigrid	multigrid	PROPN
ejpam-4835	173	20	code	code	NOUN
ejpam-4835	173	21	.	.	PUNCT
ejpam-4835	174	1	with	with	ADP
ejpam-4835	174	2	respect	respect	NOUN
ejpam-4835	174	3	to	to	ADP
ejpam-4835	174	4	recursion	recursion	NOUN
ejpam-4835	174	5	in	in	ADP
ejpam-4835	174	6	the	the	DET
ejpam-4835	174	7	multigrid	multigrid	NOUN
ejpam-4835	174	8	method	method	NOUN
ejpam-4835	174	9	,	,	PUNCT
ejpam-4835	174	10	the	the	DET
ejpam-4835	174	11	recursive	recursive	ADJ
ejpam-4835	174	12	algorithm	algorithm	NOUN
ejpam-4835	174	13	is	be	AUX
ejpam-4835	174	14	terminated	terminate	VERB
ejpam-4835	174	15	when	when	SCONJ
ejpam-4835	174	16	the	the	DET
ejpam-4835	174	17	degrees	degree	NOUN
ejpam-4835	174	18	of	of	ADP
ejpam-4835	174	19	freedom	freedom	NOUN
ejpam-4835	174	20	,	,	PUNCT
ejpam-4835	174	21	represented	represent	VERB
ejpam-4835	174	22	by	by	ADP
ejpam-4835	174	23	the	the	DET
ejpam-4835	174	24	number	number	NOUN
ejpam-4835	174	25	of	of	ADP
ejpam-4835	174	26	interior	interior	ADJ
ejpam-4835	174	27	grid	grid	NOUN
ejpam-4835	174	28	points	point	NOUN
ejpam-4835	174	29	,	,	PUNCT
ejpam-4835	174	30	become	become	VERB
ejpam-4835	174	31	less	less	ADJ
ejpam-4835	174	32	than	than	ADP
ejpam-4835	174	33	5	5	NUM
ejpam-4835	174	34	.	.	PUNCT
ejpam-4835	174	35	figure	figure	NOUN
ejpam-4835	174	36	2	2	NUM
ejpam-4835	174	37	illustrates	illustrate	VERB
ejpam-4835	174	38	the	the	DET
ejpam-4835	174	39	convergence	convergence	NOUN
ejpam-4835	174	40	behavior	behavior	NOUN
ejpam-4835	174	41	of	of	ADP
ejpam-4835	174	42	the	the	DET
ejpam-4835	174	43	figure	figure	NOUN
ejpam-4835	174	44	2	2	NUM
ejpam-4835	174	45	:	:	PUNCT
ejpam-4835	174	46	comparison	comparison	NOUN
ejpam-4835	174	47	between	between	ADP
ejpam-4835	174	48	the	the	DET
ejpam-4835	174	49	convergence	convergence	NOUN
ejpam-4835	174	50	behavior	behavior	NOUN
ejpam-4835	174	51	of	of	ADP
ejpam-4835	174	52	multigrid	multigrid	PROPN
ejpam-4835	174	53	method	method	NOUN
ejpam-4835	174	54	and	and	CCONJ
ejpam-4835	174	55	gauss	gauss	ADJ
ejpam-4835	174	56	-	-	PUNCT
ejpam-4835	174	57	seidel	seidel	NOUN
ejpam-4835	174	58	methods	method	NOUN
ejpam-4835	174	59	.	.	PUNCT
ejpam-4835	175	1	multigrid	multigrid	PROPN
ejpam-4835	175	2	solver	solver	PROPN
ejpam-4835	175	3	(	(	PUNCT
ejpam-4835	175	4	green	green	ADJ
ejpam-4835	175	5	and	and	CCONJ
ejpam-4835	175	6	red	red	ADJ
ejpam-4835	175	7	curves	curve	NOUN
ejpam-4835	175	8	for	for	ADP
ejpam-4835	175	9	the	the	DET
ejpam-4835	175	10	maximum	maximum	ADJ
ejpam-4835	175	11	norm	norm	NOUN
ejpam-4835	175	12	of	of	ADP
ejpam-4835	175	13	the	the	DET
ejpam-4835	175	14	residual	residual	NOUN
ejpam-4835	175	15	of	of	ADP
ejpam-4835	175	16	multigrid	multigrid	NOUN
ejpam-4835	175	17	(	(	PUNCT
ejpam-4835	175	18	v	v	NOUN
ejpam-4835	175	19	and	and	CCONJ
ejpam-4835	175	20	w	w	NOUN
ejpam-4835	175	21	cycle	cycle	NOUN
ejpam-4835	175	22	)	)	PUNCT
ejpam-4835	175	23	)	)	PUNCT
ejpam-4835	176	1	with	with	ADP
ejpam-4835	176	2	respect	respect	NOUN
ejpam-4835	176	3	to	to	ADP
ejpam-4835	176	4	the	the	DET
ejpam-4835	176	5	number	number	NOUN
ejpam-4835	176	6	of	of	ADP
ejpam-4835	176	7	iterations	iteration	NOUN
ejpam-4835	176	8	performed	perform	VERB
ejpam-4835	176	9	.	.	PUNCT
ejpam-4835	177	1	for	for	ADP
ejpam-4835	177	2	comparison	comparison	NOUN
ejpam-4835	177	3	,	,	PUNCT
ejpam-4835	177	4	the	the	DET
ejpam-4835	177	5	convergence	convergence	NOUN
ejpam-4835	177	6	behavior	behavior	NOUN
ejpam-4835	177	7	of	of	ADP
ejpam-4835	177	8	gauss	gauss	PROPN
ejpam-4835	177	9	-	-	PUNCT
ejpam-4835	177	10	seidel	seidel	PROPN
ejpam-4835	177	11	(	(	PUNCT
ejpam-4835	177	12	blue	blue	ADJ
ejpam-4835	177	13	curves	curve	NOUN
ejpam-4835	177	14	)	)	PUNCT
ejpam-4835	177	15	are	be	AUX
ejpam-4835	177	16	included	include	VERB
ejpam-4835	177	17	.	.	PUNCT
ejpam-4835	178	1	the	the	DET
ejpam-4835	178	2	multigrid	multigrid	NOUN
ejpam-4835	178	3	v	v	ADP
ejpam-4835	178	4	-cycle	-cycle	NOUN
ejpam-4835	178	5	is	be	AUX
ejpam-4835	178	6	carried	carry	VERB
ejpam-4835	178	7	out	out	ADP
ejpam-4835	178	8	on	on	ADP
ejpam-4835	178	9	the	the	DET
ejpam-4835	178	10	finest	fine	ADJ
ejpam-4835	178	11	grid	grid	NOUN
ejpam-4835	178	12	with	with	ADP
ejpam-4835	178	13	1089	1089	NUM
ejpam-4835	178	14	nodes	node	NOUN
ejpam-4835	178	15	and	and	CCONJ
ejpam-4835	178	16	4	4	NUM
ejpam-4835	178	17	nodes	node	NOUN
ejpam-4835	178	18	on	on	ADP
ejpam-4835	178	19	the	the	DET
ejpam-4835	178	20	coarsest	coarse	ADJ
ejpam-4835	178	21	one	one	NUM
ejpam-4835	178	22	,	,	PUNCT
ejpam-4835	178	23	and	and	CCONJ
ejpam-4835	178	24	then	then	ADV
ejpam-4835	178	25	we	we	PRON
ejpam-4835	178	26	have	have	AUX
ejpam-4835	178	27	applied	apply	VERB
ejpam-4835	178	28	thethe	thethe	ADJ
ejpam-4835	178	29	matlab	matlab	PROPN
ejpam-4835	178	30	backslash	backslash	NOUN
ejpam-4835	178	31	-	-	PUNCT
ejpam-4835	178	32	operator	operator	NOUN
ejpam-4835	178	33	and	and	CCONJ
ejpam-4835	178	34	gauss	gauss	NOUN
ejpam-4835	178	35	-	-	PUNCT
ejpam-4835	178	36	seidel	seidel	NOUN
ejpam-4835	178	37	on	on	ADP
ejpam-4835	178	38	this	this	DET
ejpam-4835	178	39	finest	fine	ADJ
ejpam-4835	178	40	grid	grid	NOUN
ejpam-4835	178	41	to	to	ADP
ejpam-4835	178	42	h.	h.	PROPN
ejpam-4835	178	43	ahmad	ahmad	PROPN
ejpam-4835	178	44	et	et	PROPN
ejpam-4835	178	45	al	al	PROPN
ejpam-4835	178	46	.	.	PUNCT
ejpam-4835	178	47	/	/	SYM
ejpam-4835	178	48	eur	eur	PROPN
ejpam-4835	178	49	.	.	PUNCT
ejpam-4835	179	1	j.	j.	PROPN
ejpam-4835	179	2	pure	pure	PROPN
ejpam-4835	179	3	appl	appl	PROPN
ejpam-4835	179	4	.	.	PROPN
ejpam-4835	179	5	math	math	PROPN
ejpam-4835	179	6	,	,	PUNCT
ejpam-4835	179	7	16	16	NUM
ejpam-4835	179	8	(	(	PUNCT
ejpam-4835	179	9	3	3	NUM
ejpam-4835	179	10	)	)	PUNCT
ejpam-4835	179	11	(	(	PUNCT
ejpam-4835	179	12	2023	2023	NUM
ejpam-4835	179	13	)	)	PUNCT
ejpam-4835	179	14	,	,	PUNCT
ejpam-4835	179	15	1956	1956	NUM
ejpam-4835	179	16	-	-	SYM
ejpam-4835	179	17	1969	1969	NUM
ejpam-4835	179	18	1965	1965	NUM
ejpam-4835	179	19	get	get	VERB
ejpam-4835	179	20	the	the	DET
ejpam-4835	179	21	solutions	solution	NOUN
ejpam-4835	179	22	in	in	ADP
ejpam-4835	179	23	figures	figure	NOUN
ejpam-4835	179	24	3	3	NUM
ejpam-4835	179	25	-	-	SYM
ejpam-4835	179	26	6.	6.	NUM
ejpam-4835	179	27	norm	norm	NOUN
ejpam-4835	179	28	of	of	ADP
ejpam-4835	179	29	residual	residual	ADJ
ejpam-4835	179	30	obtained	obtain	VERB
ejpam-4835	179	31	by	by	ADP
ejpam-4835	179	32	gauss	gauss	PROPN
ejpam-4835	179	33	seidel	seidel	PROPN
ejpam-4835	179	34	method	method	NOUN
ejpam-4835	179	35	after	after	ADP
ejpam-4835	179	36	100	100	NUM
ejpam-4835	179	37	iteratons	iteraton	NOUN
ejpam-4835	179	38	:	:	PUNCT
ejpam-4835	179	39	=	=	SYM
ejpam-4835	179	40	0.1303	0.1303	NUM
ejpam-4835	179	41	.	.	PUNCT
ejpam-4835	180	1	norm	norm	NOUN
ejpam-4835	180	2	of	of	ADP
ejpam-4835	180	3	residual	residual	ADJ
ejpam-4835	180	4	obtained	obtain	VERB
ejpam-4835	180	5	by	by	ADP
ejpam-4835	180	6	multigrid	multigrid	NOUN
ejpam-4835	180	7	v	v	NOUN
ejpam-4835	180	8	-	-	PUNCT
ejpam-4835	180	9	cycle	cycle	NOUN
ejpam-4835	180	10	after	after	ADP
ejpam-4835	180	11	100	100	NUM
ejpam-4835	180	12	iterations	iteration	NOUN
ejpam-4835	180	13	:	:	PUNCT
ejpam-4835	180	14	=	=	SYM
ejpam-4835	180	15	3.7937e−	3.7937e−	NUM
ejpam-4835	180	16	11	11	NUM
ejpam-4835	180	17	.	.	PUNCT
ejpam-4835	181	1	norm	norm	NOUN
ejpam-4835	181	2	of	of	ADP
ejpam-4835	181	3	residual	residual	ADJ
ejpam-4835	181	4	obtained	obtain	VERB
ejpam-4835	181	5	by	by	ADP
ejpam-4835	181	6	multigrid	multigrid	PROPN
ejpam-4835	181	7	w	w	NOUN
ejpam-4835	181	8	-	-	PUNCT
ejpam-4835	181	9	cycle	cycle	NOUN
ejpam-4835	181	10	after	after	ADP
ejpam-4835	181	11	100	100	NUM
ejpam-4835	181	12	iterations	iteration	NOUN
ejpam-4835	181	13	:	:	PUNCT
ejpam-4835	181	14	=	=	SYM
ejpam-4835	181	15	1.6209e−	1.6209e−	NUM
ejpam-4835	181	16	14	14	NUM
ejpam-4835	181	17	.	.	PUNCT
ejpam-4835	182	1	notting	notte	VERB
ejpam-4835	182	2	that	that	PRON
ejpam-4835	182	3	,	,	PUNCT
ejpam-4835	182	4	if	if	SCONJ
ejpam-4835	182	5	we	we	PRON
ejpam-4835	182	6	perform	perform	VERB
ejpam-4835	182	7	more	more	ADJ
ejpam-4835	182	8	than	than	ADP
ejpam-4835	182	9	20	20	NUM
ejpam-4835	182	10	iterations	iteration	NOUN
ejpam-4835	182	11	,	,	PUNCT
ejpam-4835	182	12	the	the	DET
ejpam-4835	182	13	multigrid	multigrid	NOUN
ejpam-4835	182	14	solution	solution	NOUN
ejpam-4835	182	15	is	be	AUX
ejpam-4835	182	16	better	well	ADJ
ejpam-4835	182	17	than	than	ADP
ejpam-4835	182	18	the	the	DET
ejpam-4835	182	19	matlab	matlab	PROPN
ejpam-4835	182	20	backslash	backslash	NOUN
ejpam-4835	182	21	-	-	PUNCT
ejpam-4835	182	22	operator	operator	NOUN
ejpam-4835	182	23	(	(	PUNCT
ejpam-4835	182	24	mbo	mbo	NOUN
ejpam-4835	182	25	)	)	PUNCT
ejpam-4835	182	26	solution.	solution.	PROPN
ejpam-4835	182	27	norm	norm	NOUN
ejpam-4835	182	28	of	of	ADP
ejpam-4835	182	29	residual	residual	ADJ
ejpam-4835	182	30	obtained	obtain	VERB
ejpam-4835	182	31	by	by	ADP
ejpam-4835	182	32	gauss	gauss	PROPN
ejpam-4835	182	33	seidel	seidel	PROPN
ejpam-4835	182	34	method	method	NOUN
ejpam-4835	182	35	after	after	ADP
ejpam-4835	182	36	100	100	NUM
ejpam-4835	182	37	iteratons	iteraton	NOUN
ejpam-4835	182	38	:	:	PUNCT
ejpam-4835	182	39	=	=	SYM
ejpam-4835	182	40	0.1303	0.1303	NUM
ejpam-4835	182	41	.	.	PUNCT
ejpam-4835	183	1	norm	norm	NOUN
ejpam-4835	183	2	of	of	ADP
ejpam-4835	183	3	residual	residual	ADJ
ejpam-4835	183	4	obtained	obtain	VERB
ejpam-4835	183	5	by	by	ADP
ejpam-4835	183	6	multigrid	multigrid	NOUN
ejpam-4835	183	7	v	v	NOUN
ejpam-4835	183	8	-	-	PUNCT
ejpam-4835	183	9	cycle	cycle	NOUN
ejpam-4835	183	10	after	after	ADP
ejpam-4835	183	11	20	20	NUM
ejpam-4835	183	12	iterations	iteration	NOUN
ejpam-4835	183	13	:	:	PUNCT
ejpam-4835	183	14	=	=	SYM
ejpam-4835	183	15	0.0049	0.0049	NUM
ejpam-4835	183	16	.	.	PUNCT
ejpam-4835	183	17	norm	norm	NOUN
ejpam-4835	183	18	of	of	ADP
ejpam-4835	183	19	residual	residual	ADJ
ejpam-4835	183	20	obtained	obtain	VERB
ejpam-4835	183	21	by	by	ADP
ejpam-4835	183	22	multigrid	multigrid	PROPN
ejpam-4835	183	23	w	w	NOUN
ejpam-4835	183	24	-	-	PUNCT
ejpam-4835	183	25	cycle	cycle	NOUN
ejpam-4835	183	26	after	after	ADP
ejpam-4835	183	27	20	20	NUM
ejpam-4835	183	28	iterations	iteration	NOUN
ejpam-4835	183	29	:	:	PUNCT
ejpam-4835	183	30	=	=	SYM
ejpam-4835	183	31	4.5777e−	4.5777e−	NUM
ejpam-4835	183	32	5	5	NUM
ejpam-4835	183	33	.	.	PUNCT
ejpam-4835	183	34	figure	figure	VERB
ejpam-4835	183	35	3	3	NUM
ejpam-4835	183	36	:	:	PUNCT
ejpam-4835	183	37	solution	solution	NOUN
ejpam-4835	183	38	of	of	ADP
ejpam-4835	183	39	the	the	DET
ejpam-4835	183	40	problem	problem	NOUN
ejpam-4835	183	41	4.1	4.1	NUM
ejpam-4835	183	42	on	on	ADP
ejpam-4835	183	43	fine	fine	ADJ
ejpam-4835	183	44	grid	grid	NOUN
ejpam-4835	183	45	with	with	ADP
ejpam-4835	183	46	1089	1089	NUM
ejpam-4835	183	47	dofs	dof	NOUN
ejpam-4835	183	48	using	use	VERB
ejpam-4835	183	49	matlab	matlab	PROPN
ejpam-4835	183	50	backslash	backslash	NOUN
ejpam-4835	183	51	operator	operator	NOUN
ejpam-4835	183	52	solver	solver	VERB
ejpam-4835	183	53	after	after	ADP
ejpam-4835	183	54	100	100	NUM
ejpam-4835	183	55	iterations	iteration	NOUN
ejpam-4835	183	56	.	.	PUNCT
ejpam-4835	184	1	figure	figure	VERB
ejpam-4835	184	2	4	4	NUM
ejpam-4835	184	3	:	:	PUNCT
ejpam-4835	184	4	solution	solution	NOUN
ejpam-4835	184	5	of	of	ADP
ejpam-4835	184	6	the	the	DET
ejpam-4835	184	7	problem	problem	NOUN
ejpam-4835	184	8	4.1	4.1	NUM
ejpam-4835	184	9	on	on	ADP
ejpam-4835	184	10	fine	fine	ADJ
ejpam-4835	184	11	grid	grid	NOUN
ejpam-4835	184	12	with	with	ADP
ejpam-4835	184	13	1089	1089	NUM
ejpam-4835	184	14	dofs	dof	NOUN
ejpam-4835	184	15	using	use	VERB
ejpam-4835	184	16	gauss	gauss	PROPN
ejpam-4835	184	17	seidel	seidel	PROPN
ejpam-4835	184	18	method	method	NOUN
ejpam-4835	184	19	after	after	ADP
ejpam-4835	184	20	100	100	NUM
ejpam-4835	184	21	iterations	iteration	NOUN
ejpam-4835	184	22	.	.	PUNCT
ejpam-4835	185	1	5	5	X
ejpam-4835	185	2	.	.	X
ejpam-4835	185	3	conclusion	conclusion	NOUN
ejpam-4835	185	4	in	in	ADP
ejpam-4835	185	5	this	this	DET
ejpam-4835	185	6	study	study	NOUN
ejpam-4835	185	7	,	,	PUNCT
ejpam-4835	185	8	we	we	PRON
ejpam-4835	185	9	utilize	utilize	VERB
ejpam-4835	185	10	the	the	DET
ejpam-4835	185	11	algebraic	algebraic	ADJ
ejpam-4835	185	12	multigrid	multigrid	NOUN
ejpam-4835	185	13	method	method	NOUN
ejpam-4835	185	14	,	,	PUNCT
ejpam-4835	185	15	specifically	specifically	ADV
ejpam-4835	185	16	the	the	DET
ejpam-4835	185	17	efficient	efficient	ADJ
ejpam-4835	185	18	iterative	iterative	NOUN
ejpam-4835	185	19	solutions	solution	NOUN
ejpam-4835	185	20	for	for	ADP
ejpam-4835	185	21	discretizing	discretize	VERB
ejpam-4835	185	22	elliptic	elliptic	ADJ
ejpam-4835	185	23	variational	variational	ADJ
ejpam-4835	185	24	inequalities	inequality	NOUN
ejpam-4835	185	25	,	,	PUNCT
ejpam-4835	185	26	to	to	PART
ejpam-4835	185	27	handle	handle	VERB
ejpam-4835	185	28	the	the	DET
ejpam-4835	185	29	discretization	discretization	NOUN
ejpam-4835	185	30	h.	h.	PROPN
ejpam-4835	185	31	ahmad	ahmad	PROPN
ejpam-4835	185	32	et	et	PROPN
ejpam-4835	185	33	al	al	PROPN
ejpam-4835	185	34	.	.	PUNCT
ejpam-4835	185	35	/	/	SYM
ejpam-4835	185	36	eur	eur	PROPN
ejpam-4835	185	37	.	.	PUNCT
ejpam-4835	186	1	j.	j.	PROPN
ejpam-4835	186	2	pure	pure	PROPN
ejpam-4835	186	3	appl	appl	PROPN
ejpam-4835	186	4	.	.	PROPN
ejpam-4835	186	5	math	math	PROPN
ejpam-4835	186	6	,	,	PUNCT
ejpam-4835	186	7	16	16	NUM
ejpam-4835	186	8	(	(	PUNCT
ejpam-4835	186	9	3	3	NUM
ejpam-4835	186	10	)	)	PUNCT
ejpam-4835	186	11	(	(	PUNCT
ejpam-4835	186	12	2023	2023	NUM
ejpam-4835	186	13	)	)	PUNCT
ejpam-4835	186	14	,	,	PUNCT
ejpam-4835	186	15	1956	1956	NUM
ejpam-4835	186	16	-	-	SYM
ejpam-4835	186	17	1969	1969	NUM
ejpam-4835	186	18	1966	1966	NUM
ejpam-4835	186	19	figure	figure	NOUN
ejpam-4835	186	20	5	5	NUM
ejpam-4835	186	21	:	:	PUNCT
ejpam-4835	186	22	solution	solution	NOUN
ejpam-4835	186	23	of	of	ADP
ejpam-4835	186	24	the	the	DET
ejpam-4835	186	25	problem	problem	NOUN
ejpam-4835	186	26	4.1	4.1	NUM
ejpam-4835	186	27	on	on	ADP
ejpam-4835	186	28	fine	fine	ADJ
ejpam-4835	186	29	grid	grid	NOUN
ejpam-4835	186	30	with	with	ADP
ejpam-4835	186	31	1089	1089	NUM
ejpam-4835	186	32	dofs	dof	NOUN
ejpam-4835	186	33	using	use	VERB
ejpam-4835	186	34	multigrid	multigrid	NOUN
ejpam-4835	186	35	method	method	NOUN
ejpam-4835	186	36	v	v	NOUN
ejpam-4835	186	37	-	-	PUNCT
ejpam-4835	186	38	cycle	cycle	NOUN
ejpam-4835	186	39	after	after	ADP
ejpam-4835	186	40	100	100	NUM
ejpam-4835	186	41	iterations	iteration	NOUN
ejpam-4835	186	42	.	.	PUNCT
ejpam-4835	187	1	figure	figure	VERB
ejpam-4835	187	2	6	6	NUM
ejpam-4835	187	3	:	:	PUNCT
ejpam-4835	187	4	solution	solution	NOUN
ejpam-4835	187	5	of	of	ADP
ejpam-4835	187	6	the	the	DET
ejpam-4835	187	7	problem	problem	NOUN
ejpam-4835	187	8	4.1	4.1	NUM
ejpam-4835	187	9	on	on	ADP
ejpam-4835	187	10	fine	fine	ADJ
ejpam-4835	187	11	grid	grid	NOUN
ejpam-4835	187	12	with	with	ADP
ejpam-4835	187	13	1089	1089	NUM
ejpam-4835	187	14	dofs	dof	NOUN
ejpam-4835	187	15	using	use	VERB
ejpam-4835	187	16	multigrid	multigrid	NOUN
ejpam-4835	187	17	method	method	NOUN
ejpam-4835	187	18	w	w	NOUN
ejpam-4835	187	19	-	-	PUNCT
ejpam-4835	187	20	cycle	cycle	NOUN
ejpam-4835	187	21	after	after	ADP
ejpam-4835	187	22	100	100	NUM
ejpam-4835	187	23	iterations	iteration	NOUN
ejpam-4835	187	24	.	.	PUNCT
ejpam-4835	188	1	of	of	ADP
ejpam-4835	188	2	a	a	DET
ejpam-4835	188	3	loop	loop	NOUN
ejpam-4835	188	4	domain	domain	NOUN
ejpam-4835	188	5	using	use	VERB
ejpam-4835	188	6	adaptive	adaptive	ADJ
ejpam-4835	188	7	finite	finite	ADJ
ejpam-4835	188	8	element	element	NOUN
ejpam-4835	188	9	approximation	approximation	NOUN
ejpam-4835	188	10	.	.	PUNCT
ejpam-4835	189	1	after	after	ADP
ejpam-4835	189	2	discretization	discretization	NOUN
ejpam-4835	189	3	,	,	PUNCT
ejpam-4835	189	4	we	we	PRON
ejpam-4835	189	5	implement	implement	VERB
ejpam-4835	189	6	the	the	DET
ejpam-4835	189	7	multigrid	multigrid	NOUN
ejpam-4835	189	8	method	method	NOUN
ejpam-4835	189	9	to	to	PART
ejpam-4835	189	10	efficiently	efficiently	ADV
ejpam-4835	189	11	solve	solve	VERB
ejpam-4835	189	12	the	the	DET
ejpam-4835	189	13	discrete	discrete	ADJ
ejpam-4835	189	14	problems	problem	NOUN
ejpam-4835	189	15	.	.	PUNCT
ejpam-4835	190	1	we	we	PRON
ejpam-4835	190	2	investigate	investigate	VERB
ejpam-4835	190	3	the	the	DET
ejpam-4835	190	4	uniform	uniform	ADJ
ejpam-4835	190	5	convergence	convergence	NOUN
ejpam-4835	190	6	of	of	ADP
ejpam-4835	190	7	our	our	PRON
ejpam-4835	190	8	approach	approach	NOUN
ejpam-4835	190	9	and	and	CCONJ
ejpam-4835	190	10	demonstrate	demonstrate	VERB
ejpam-4835	190	11	that	that	SCONJ
ejpam-4835	190	12	the	the	DET
ejpam-4835	190	13	multigrid	multigrid	NOUN
ejpam-4835	190	14	method	method	NOUN
ejpam-4835	190	15	exhibits	exhibit	VERB
ejpam-4835	190	16	a	a	DET
ejpam-4835	190	17	significant	significant	ADJ
ejpam-4835	190	18	reduction	reduction	NOUN
ejpam-4835	190	19	in	in	ADP
ejpam-4835	190	20	the	the	DET
ejpam-4835	190	21	number	number	NOUN
ejpam-4835	190	22	of	of	ADP
ejpam-4835	190	23	iterations	iteration	NOUN
ejpam-4835	190	24	compared	compare	VERB
ejpam-4835	190	25	to	to	ADP
ejpam-4835	190	26	the	the	DET
ejpam-4835	190	27	maximum	maximum	ADJ
ejpam-4835	190	28	norm	norm	NOUN
ejpam-4835	190	29	.	.	PUNCT
ejpam-4835	191	1	in	in	ADP
ejpam-4835	191	2	our	our	PRON
ejpam-4835	191	3	numerical	numerical	ADJ
ejpam-4835	191	4	experiments	experiment	NOUN
ejpam-4835	191	5	,	,	PUNCT
ejpam-4835	191	6	we	we	PRON
ejpam-4835	191	7	present	present	VERB
ejpam-4835	191	8	an	an	DET
ejpam-4835	191	9	example	example	NOUN
ejpam-4835	191	10	of	of	ADP
ejpam-4835	191	11	a	a	DET
ejpam-4835	191	12	variational	variational	ADJ
ejpam-4835	191	13	inequality	inequality	NOUN
ejpam-4835	191	14	.	.	PUNCT
ejpam-4835	192	1	the	the	DET
ejpam-4835	192	2	results	result	NOUN
ejpam-4835	192	3	indicate	indicate	VERB
ejpam-4835	192	4	that	that	SCONJ
ejpam-4835	192	5	the	the	DET
ejpam-4835	192	6	gauss	gauss	PROPN
ejpam-4835	192	7	-	-	PUNCT
ejpam-4835	192	8	seidel	seidel	PROPN
ejpam-4835	192	9	method	method	NOUN
ejpam-4835	192	10	does	do	AUX
ejpam-4835	192	11	not	not	PART
ejpam-4835	192	12	perform	perform	VERB
ejpam-4835	192	13	well	well	ADV
ejpam-4835	192	14	even	even	ADV
ejpam-4835	192	15	after	after	ADP
ejpam-4835	192	16	a	a	DET
ejpam-4835	192	17	large	large	ADJ
ejpam-4835	192	18	number	number	NOUN
ejpam-4835	192	19	of	of	ADP
ejpam-4835	192	20	iterations	iteration	NOUN
ejpam-4835	192	21	.	.	PUNCT
ejpam-4835	193	1	in	in	ADP
ejpam-4835	193	2	contrast	contrast	NOUN
ejpam-4835	193	3	,	,	PUNCT
ejpam-4835	193	4	the	the	DET
ejpam-4835	193	5	multigrid	multigrid	NOUN
ejpam-4835	193	6	method	method	NOUN
ejpam-4835	193	7	,	,	PUNCT
ejpam-4835	193	8	with	with	ADP
ejpam-4835	193	9	its	its	PRON
ejpam-4835	193	10	debug	debug	NOUN
ejpam-4835	193	11	function	function	NOUN
ejpam-4835	193	12	that	that	PRON
ejpam-4835	193	13	reduces	reduce	VERB
ejpam-4835	193	14	high	high	ADJ
ejpam-4835	193	15	-	-	PUNCT
ejpam-4835	193	16	frequency	frequency	NOUN
ejpam-4835	193	17	errors	error	NOUN
ejpam-4835	193	18	through	through	ADP
ejpam-4835	193	19	relaxation	relaxation	NOUN
ejpam-4835	193	20	and	and	CCONJ
ejpam-4835	193	21	maps	map	VERB
ejpam-4835	193	22	low	low	ADJ
ejpam-4835	193	23	-	-	PUNCT
ejpam-4835	193	24	frequency	frequency	NOUN
ejpam-4835	193	25	errors	error	NOUN
ejpam-4835	193	26	to	to	ADP
ejpam-4835	193	27	the	the	DET
ejpam-4835	193	28	coarse	coarse	ADJ
ejpam-4835	193	29	grid	grid	NOUN
ejpam-4835	193	30	for	for	ADP
ejpam-4835	193	31	reduction	reduction	NOUN
ejpam-4835	193	32	,	,	PUNCT
ejpam-4835	193	33	achieves	achieve	VERB
ejpam-4835	193	34	convergence	convergence	NOUN
ejpam-4835	193	35	in	in	ADP
ejpam-4835	193	36	a	a	DET
ejpam-4835	193	37	small	small	ADJ
ejpam-4835	193	38	number	number	NOUN
ejpam-4835	193	39	of	of	ADP
ejpam-4835	193	40	iterations	iteration	NOUN
ejpam-4835	193	41	.	.	PUNCT
ejpam-4835	194	1	we	we	PRON
ejpam-4835	194	2	acknowledge	acknowledge	VERB
ejpam-4835	194	3	the	the	DET
ejpam-4835	194	4	potential	potential	NOUN
ejpam-4835	194	5	for	for	ADP
ejpam-4835	194	6	many	many	ADJ
ejpam-4835	194	7	extensions	extension	NOUN
ejpam-4835	194	8	of	of	ADP
ejpam-4835	194	9	these	these	DET
ejpam-4835	194	10	techniques	technique	NOUN
ejpam-4835	194	11	.	.	PUNCT
ejpam-4835	195	1	an	an	DET
ejpam-4835	195	2	interesting	interesting	ADJ
ejpam-4835	195	3	future	future	ADJ
ejpam-4835	195	4	direction	direction	NOUN
ejpam-4835	195	5	could	could	AUX
ejpam-4835	195	6	involve	involve	VERB
ejpam-4835	195	7	applying	apply	VERB
ejpam-4835	195	8	a	a	DET
ejpam-4835	195	9	parallel	parallel	ADJ
ejpam-4835	195	10	full	full	ADJ
ejpam-4835	195	11	multigrid	multigrid	NOUN
ejpam-4835	195	12	method	method	NOUN
ejpam-4835	195	13	to	to	PART
ejpam-4835	195	14	solve	solve	VERB
ejpam-4835	195	15	unconstrained	unconstrained	ADJ
ejpam-4835	195	16	elliptical	elliptical	ADJ
ejpam-4835	195	17	inequalities	inequality	NOUN
ejpam-4835	195	18	.	.	PUNCT
ejpam-4835	196	1	this	this	DET
ejpam-4835	196	2	approach	approach	NOUN
ejpam-4835	196	3	may	may	AUX
ejpam-4835	196	4	further	far	ADV
ejpam-4835	196	5	enhance	enhance	VERB
ejpam-4835	196	6	the	the	DET
ejpam-4835	196	7	efficiency	efficiency	NOUN
ejpam-4835	196	8	and	and	CCONJ
ejpam-4835	196	9	scalability	scalability	NOUN
ejpam-4835	196	10	of	of	ADP
ejpam-4835	196	11	our	our	PRON
ejpam-4835	196	12	numerical	numerical	ADJ
ejpam-4835	196	13	solution	solution	NOUN
ejpam-4835	196	14	for	for	ADP
ejpam-4835	196	15	a	a	DET
ejpam-4835	196	16	broader	broad	ADJ
ejpam-4835	196	17	range	range	NOUN
ejpam-4835	196	18	of	of	ADP
ejpam-4835	196	19	problems	problem	NOUN
ejpam-4835	196	20	.	.	PUNCT
ejpam-4835	197	1	acknowledgements	acknowledgement	NOUN
ejpam-4835	197	2	this	this	DET
ejpam-4835	197	3	project	project	NOUN
ejpam-4835	197	4	is	be	AUX
ejpam-4835	197	5	funded	fund	VERB
ejpam-4835	197	6	by	by	ADP
ejpam-4835	197	7	king	king	PROPN
ejpam-4835	197	8	saud	saud	PROPN
ejpam-4835	197	9	university	university	PROPN
ejpam-4835	197	10	,	,	PUNCT
ejpam-4835	197	11	riyadh	riyadh	PROPN
ejpam-4835	197	12	,	,	PUNCT
ejpam-4835	197	13	saudi	saudi	PROPN
ejpam-4835	197	14	arabia	arabia	PROPN
ejpam-4835	197	15	.	.	PUNCT
ejpam-4835	198	1	research	research	NOUN
ejpam-4835	198	2	supporting	support	VERB
ejpam-4835	198	3	project	project	NOUN
ejpam-4835	198	4	number	number	NOUN
ejpam-4835	198	5	(	(	PUNCT
ejpam-4835	198	6	rsp2023r167	rsp2023r167	ADJ
ejpam-4835	198	7	)	)	PUNCT
ejpam-4835	198	8	,	,	PUNCT
ejpam-4835	198	9	king	king	PROPN
ejpam-4835	198	10	saud	saud	PROPN
ejpam-4835	198	11	university	university	PROPN
ejpam-4835	198	12	,	,	PUNCT
ejpam-4835	198	13	riyadh	riyadh	PROPN
ejpam-4835	198	14	,	,	PUNCT
ejpam-4835	198	15	saudi	saudi	PROPN
ejpam-4835	198	16	arabia	arabia	PROPN
ejpam-4835	198	17	.	.	PUNCT
ejpam-4835	199	1	references	reference	NOUN
ejpam-4835	199	2	1967	1967	NUM
ejpam-4835	199	3	references	reference	NOUN
ejpam-4835	199	4	[	[	X
ejpam-4835	199	5	1	1	NUM
ejpam-4835	199	6	]	]	X
ejpam-4835	199	7	m	m	VERB
ejpam-4835	199	8	adel	adel	PROPN
ejpam-4835	199	9	,	,	PUNCT
ejpam-4835	199	10	nh	nh	PROPN
ejpam-4835	199	11	sweilam	sweilam	PROPN
ejpam-4835	199	12	,	,	PUNCT
ejpam-4835	199	13	mmkhader	mmkhader	PROPN
ejpam-4835	199	14	,	,	PUNCT
ejpam-4835	199	15	sm	sm	PROPN
ejpam-4835	199	16	ahmed	ahmed	PROPN
ejpam-4835	199	17	,	,	PUNCT
ejpam-4835	199	18	h	h	PROPN
ejpam-4835	199	19	ahmad	ahmad	PROPN
ejpam-4835	199	20	,	,	PUNCT
ejpam-4835	199	21	and	and	CCONJ
ejpam-4835	199	22	t	t	PROPN
ejpam-4835	199	23	botmart	botmart	NOUN
ejpam-4835	199	24	.	.	PUNCT
ejpam-4835	200	1	numerical	numerical	PROPN
ejpam-4835	200	2	simulation	simulation	PROPN
ejpam-4835	200	3	using	use	VERB
ejpam-4835	200	4	the	the	DET
ejpam-4835	200	5	non	non	ADJ
ejpam-4835	200	6	-	-	ADJ
ejpam-4835	200	7	standard	standard	ADJ
ejpam-4835	200	8	weighted	weight	VERB
ejpam-4835	200	9	average	average	ADJ
ejpam-4835	200	10	fdm	fdm	NOUN
ejpam-4835	200	11	for	for	ADP
ejpam-4835	200	12	2dim	2dim	PROPN
ejpam-4835	200	13	variable	variable	ADJ
ejpam-4835	200	14	-	-	PUNCT
ejpam-4835	200	15	order	order	NOUN
ejpam-4835	200	16	cable	cable	NOUN
ejpam-4835	200	17	equation	equation	NOUN
ejpam-4835	200	18	.	.	PUNCT
ejpam-4835	201	1	results	result	NOUN
ejpam-4835	201	2	in	in	ADP
ejpam-4835	201	3	physics	physics	NOUN
ejpam-4835	201	4	,	,	PUNCT
ejpam-4835	201	5	39:105682	39:105682	NOUN
ejpam-4835	201	6	,	,	PUNCT
ejpam-4835	201	7	2022	2022	NUM
ejpam-4835	201	8	.	.	PUNCT
ejpam-4835	202	1	[	[	X
ejpam-4835	202	2	2	2	NUM
ejpam-4835	202	3	]	]	PUNCT
ejpam-4835	202	4	h	h	PROPN
ejpam-4835	202	5	ahmad	ahmad	PROPN
ejpam-4835	202	6	ae	ae	PROPN
ejpam-4835	202	7	abouelregal	abouelregal	PROPN
ejpam-4835	202	8	and	and	CCONJ
ejpam-4835	202	9	s	s	NOUN
ejpam-4835	202	10	-	-	PROPN
ejpam-4835	202	11	w	w	NOUN
ejpam-4835	202	12	yao	yao	NOUN
ejpam-4835	202	13	.	.	PUNCT
ejpam-4835	203	1	functionally	functionally	ADV
ejpam-4835	203	2	graded	grade	VERB
ejpam-4835	203	3	piezoelectric	piezoelectric	ADJ
ejpam-4835	203	4	medium	medium	NOUN
ejpam-4835	203	5	exposed	expose	VERB
ejpam-4835	203	6	to	to	ADP
ejpam-4835	203	7	a	a	DET
ejpam-4835	203	8	movable	movable	ADJ
ejpam-4835	203	9	heat	heat	NOUN
ejpam-4835	203	10	flow	flow	NOUN
ejpam-4835	203	11	based	base	VERB
ejpam-4835	203	12	on	on	ADP
ejpam-4835	203	13	a	a	DET
ejpam-4835	203	14	heat	heat	NOUN
ejpam-4835	203	15	equation	equation	NOUN
ejpam-4835	203	16	with	with	ADP
ejpam-4835	203	17	a	a	DET
ejpam-4835	203	18	memory	memory	NOUN
ejpam-4835	203	19	-	-	PUNCT
ejpam-4835	203	20	dependent	dependent	ADJ
ejpam-4835	203	21	derivative	derivative	NOUN
ejpam-4835	203	22	.	.	PUNCT
ejpam-4835	204	1	materials	material	NOUN
ejpam-4835	204	2	,	,	PUNCT
ejpam-4835	204	3	13(18):3953	13(18):3953	NUM
ejpam-4835	204	4	,	,	PUNCT
ejpam-4835	204	5	2020	2020	NUM
ejpam-4835	204	6	.	.	PUNCT
ejpam-4835	205	1	[	[	X
ejpam-4835	205	2	3	3	NUM
ejpam-4835	205	3	]	]	X
ejpam-4835	205	4	h	h	NOUN
ejpam-4835	205	5	ahmad	ahmad	PROPN
ejpam-4835	205	6	,	,	PUNCT
ejpam-4835	205	7	,	,	PUNCT
ejpam-4835	205	8	and	and	CCONJ
ejpam-4835	205	9	ta	ta	ADP
ejpam-4835	205	10	khan	khan	PROPN
ejpam-4835	205	11	.	.	PUNCT
ejpam-4835	206	1	variational	variational	ADJ
ejpam-4835	206	2	iteration	iteration	NOUN
ejpam-4835	206	3	algorithm	algorithm	NOUN
ejpam-4835	206	4	i	i	PRON
ejpam-4835	206	5	with	with	ADP
ejpam-4835	206	6	an	an	DET
ejpam-4835	206	7	auxiliary	auxiliary	ADJ
ejpam-4835	206	8	parameter	parameter	NOUN
ejpam-4835	206	9	for	for	ADP
ejpam-4835	206	10	the	the	DET
ejpam-4835	206	11	solution	solution	NOUN
ejpam-4835	206	12	of	of	ADP
ejpam-4835	206	13	differential	differential	ADJ
ejpam-4835	206	14	equations	equation	NOUN
ejpam-4835	206	15	of	of	ADP
ejpam-4835	206	16	motion	motion	NOUN
ejpam-4835	206	17	for	for	ADP
ejpam-4835	206	18	simple	simple	ADJ
ejpam-4835	206	19	and	and	CCONJ
ejpam-4835	206	20	damped	damped	ADJ
ejpam-4835	206	21	mass	mass	ADJ
ejpam-4835	206	22	–	–	PUNCT
ejpam-4835	206	23	spring	spring	NOUN
ejpam-4835	206	24	systems	system	NOUN
ejpam-4835	206	25	.	.	PUNCT
ejpam-4835	207	1	noise	noise	NOUN
ejpam-4835	207	2	&	&	CCONJ
ejpam-4835	207	3	vibration	vibration	NOUN
ejpam-4835	207	4	worldwide	worldwide	ADV
ejpam-4835	207	5	,	,	PUNCT
ejpam-4835	207	6	51(1	51(1	NOUN
ejpam-4835	207	7	-	-	PUNCT
ejpam-4835	207	8	2):12–20	2):12–20	NUM
ejpam-4835	207	9	,	,	PUNCT
ejpam-4835	207	10	2020	2020	NUM
ejpam-4835	207	11	.	.	PUNCT
ejpam-4835	208	1	[	[	X
ejpam-4835	208	2	4	4	NUM
ejpam-4835	208	3	]	]	X
ejpam-4835	208	4	h	h	PROPN
ejpam-4835	208	5	ahmad	ahmad	PROPN
ejpam-4835	208	6	,	,	PUNCT
ejpam-4835	208	7	ta	ta	PROPN
ejpam-4835	208	8	khan	khan	PROPN
ejpam-4835	208	9	,	,	PUNCT
ejpam-4835	208	10	and	and	CCONJ
ejpam-4835	208	11	s	s	X
ejpam-4835	208	12	-	-	PROPN
ejpam-4835	208	13	w	w	NOUN
ejpam-4835	208	14	yao	yao	NOUN
ejpam-4835	208	15	.	.	PUNCT
ejpam-4835	209	1	an	an	DET
ejpam-4835	209	2	efficient	efficient	ADJ
ejpam-4835	209	3	approach	approach	NOUN
ejpam-4835	209	4	for	for	ADP
ejpam-4835	209	5	the	the	DET
ejpam-4835	209	6	numerical	numerical	ADJ
ejpam-4835	209	7	solution	solution	NOUN
ejpam-4835	209	8	of	of	ADP
ejpam-4835	209	9	fifth	fifth	ADJ
ejpam-4835	209	10	-	-	PUNCT
ejpam-4835	209	11	order	order	NOUN
ejpam-4835	209	12	kdv	kdv	NOUN
ejpam-4835	209	13	equations	equation	NOUN
ejpam-4835	209	14	.	.	PUNCT
ejpam-4835	210	1	open	open	ADJ
ejpam-4835	210	2	mathematics	mathematic	NOUN
ejpam-4835	210	3	,	,	PUNCT
ejpam-4835	210	4	18(1):738–748	18(1):738–748	PROPN
ejpam-4835	210	5	,	,	PUNCT
ejpam-4835	210	6	2020	2020	NUM
ejpam-4835	210	7	.	.	PUNCT
ejpam-4835	211	1	[	[	X
ejpam-4835	211	2	5	5	NUM
ejpam-4835	211	3	]	]	X
ejpam-4835	211	4	i	i	PROPN
ejpam-4835	211	5	ahmad	ahmad	PROPN
ejpam-4835	211	6	,	,	PUNCT
ejpam-4835	211	7	ar	ar	PROPN
ejpam-4835	211	8	seadawy	seadawy	PROPN
ejpam-4835	211	9	,	,	PUNCT
ejpam-4835	211	10	h	h	PROPN
ejpam-4835	211	11	ahmad	ahmad	PROPN
ejpam-4835	211	12	,	,	PUNCT
ejpam-4835	211	13	p	p	NOUN
ejpam-4835	211	14	thounthong	thounthong	NOUN
ejpam-4835	211	15	,	,	PUNCT
ejpam-4835	211	16	and	and	CCONJ
ejpam-4835	211	17	f	f	PROPN
ejpam-4835	211	18	wang	wang	PROPN
ejpam-4835	211	19	.	.	PUNCT
ejpam-4835	212	1	numerical	numerical	PROPN
ejpam-4835	212	2	study	study	PROPN
ejpam-4835	212	3	of	of	ADP
ejpam-4835	212	4	multi	multi	ADJ
ejpam-4835	212	5	-	-	ADJ
ejpam-4835	212	6	dimensional	dimensional	ADJ
ejpam-4835	212	7	hyperbolic	hyperbolic	ADJ
ejpam-4835	212	8	telegraph	telegraph	NOUN
ejpam-4835	212	9	equations	equation	NOUN
ejpam-4835	212	10	arising	arise	VERB
ejpam-4835	212	11	in	in	ADP
ejpam-4835	212	12	nuclear	nuclear	ADJ
ejpam-4835	212	13	material	material	NOUN
ejpam-4835	212	14	science	science	NOUN
ejpam-4835	212	15	via	via	ADP
ejpam-4835	212	16	an	an	DET
ejpam-4835	212	17	efficient	efficient	ADJ
ejpam-4835	212	18	local	local	ADJ
ejpam-4835	212	19	meshless	meshless	ADJ
ejpam-4835	212	20	method	method	NOUN
ejpam-4835	212	21	.	.	PUNCT
ejpam-4835	213	1	international	international	ADJ
ejpam-4835	213	2	journal	journal	PROPN
ejpam-4835	213	3	of	of	ADP
ejpam-4835	213	4	nonlinear	nonlinear	PROPN
ejpam-4835	213	5	sciences	sciences	PROPN
ejpam-4835	213	6	and	and	CCONJ
ejpam-4835	213	7	numerical	numerical	PROPN
ejpam-4835	213	8	simulation	simulation	PROPN
ejpam-4835	213	9	,	,	PUNCT
ejpam-4835	213	10	23(1):115–122	23(1):115–122	PROPN
ejpam-4835	213	11	,	,	PUNCT
ejpam-4835	213	12	2022	2022	NUM
ejpam-4835	213	13	.	.	PUNCT
ejpam-4835	214	1	[	[	X
ejpam-4835	214	2	6	6	NUM
ejpam-4835	214	3	]	]	PUNCT
ejpam-4835	214	4	m	m	VERB
ejpam-4835	214	5	ahsan	ahsan	ADJ
ejpam-4835	214	6	,	,	PUNCT
ejpam-4835	214	7	aa	aa	PROPN
ejpam-4835	214	8	khan	khan	PROPN
ejpam-4835	214	9	,	,	PUNCT
ejpam-4835	214	10	s	s	PART
ejpam-4835	214	11	dinibutun	dinibutun	PROPN
ejpam-4835	214	12	,	,	PUNCT
ejpam-4835	214	13	i	i	PRON
ejpam-4835	214	14	ahmad	ahmad	PROPN
ejpam-4835	214	15	,	,	PUNCT
ejpam-4835	214	16	h	h	PROPN
ejpam-4835	214	17	ahmad	ahmad	PROPN
ejpam-4835	214	18	,	,	PUNCT
ejpam-4835	214	19	n	n	PRON
ejpam-4835	214	20	jarasthitikulchai	jarasthitikulchai	ADJ
ejpam-4835	214	21	,	,	PUNCT
ejpam-4835	214	22	and	and	CCONJ
ejpam-4835	214	23	w	w	NOUN
ejpam-4835	214	24	sudsutad	sudsutad	NOUN
ejpam-4835	214	25	.	.	PUNCT
ejpam-4835	215	1	the	the	DET
ejpam-4835	215	2	haar	haar	PROPN
ejpam-4835	215	3	wavelets	wavelets	PROPN
ejpam-4835	215	4	based	base	VERB
ejpam-4835	215	5	numerical	numerical	ADJ
ejpam-4835	215	6	solution	solution	NOUN
ejpam-4835	215	7	of	of	ADP
ejpam-4835	215	8	reccati	reccati	ADJ
ejpam-4835	215	9	equation	equation	NOUN
ejpam-4835	215	10	with	with	ADP
ejpam-4835	215	11	integral	integral	ADJ
ejpam-4835	215	12	boundary	boundary	ADJ
ejpam-4835	215	13	condition	condition	NOUN
ejpam-4835	215	14	.	.	PUNCT
ejpam-4835	216	1	thermal	thermal	ADJ
ejpam-4835	216	2	science	science	NOUN
ejpam-4835	216	3	,	,	PUNCT
ejpam-4835	216	4	27(spec	27(spec	NUM
ejpam-4835	216	5	.	.	PUNCT
ejpam-4835	217	1	issue	issue	NOUN
ejpam-4835	217	2	1):93–100	1):93–100	NUM
ejpam-4835	217	3	,	,	PUNCT
ejpam-4835	217	4	2023	2023	NUM
ejpam-4835	217	5	.	.	PUNCT
ejpam-4835	218	1	[	[	X
ejpam-4835	218	2	7	7	X
ejpam-4835	218	3	]	]	X
ejpam-4835	218	4	a	a	DET
ejpam-4835	218	5	akgül	akgül	PROPN
ejpam-4835	218	6	and	and	CCONJ
ejpam-4835	218	7	h	h	PROPN
ejpam-4835	218	8	ahmad	ahmad	PROPN
ejpam-4835	218	9	.	.	PUNCT
ejpam-4835	219	1	reproducing	reproduce	VERB
ejpam-4835	219	2	kernel	kernel	PROPN
ejpam-4835	219	3	method	method	NOUN
ejpam-4835	219	4	for	for	ADP
ejpam-4835	219	5	fangzhu	fangzhu	PROPN
ejpam-4835	219	6	’s	’s	PART
ejpam-4835	219	7	oscillator	oscillator	NOUN
ejpam-4835	219	8	for	for	ADP
ejpam-4835	219	9	water	water	NOUN
ejpam-4835	219	10	collection	collection	NOUN
ejpam-4835	219	11	from	from	ADP
ejpam-4835	219	12	air	air	NOUN
ejpam-4835	219	13	.	.	PUNCT
ejpam-4835	220	1	mathematical	mathematical	ADJ
ejpam-4835	220	2	methods	method	NOUN
ejpam-4835	220	3	in	in	ADP
ejpam-4835	220	4	the	the	DET
ejpam-4835	220	5	applied	apply	VERB
ejpam-4835	220	6	sciences	science	NOUN
ejpam-4835	220	7	,	,	PUNCT
ejpam-4835	220	8	2020	2020	NUM
ejpam-4835	220	9	.	.	PUNCT
ejpam-4835	221	1	[	[	X
ejpam-4835	221	2	8	8	NUM
ejpam-4835	221	3	]	]	X
ejpam-4835	221	4	r	r	NOUN
ejpam-4835	221	5	alharbi	alharbi	NOUN
ejpam-4835	221	6	,	,	PUNCT
ejpam-4835	221	7	r	r	PROPN
ejpam-4835	221	8	jan	jan	PROPN
ejpam-4835	221	9	,	,	PUNCT
ejpam-4835	221	10	s	s	PART
ejpam-4835	221	11	alyobi	alyobi	NOUN
ejpam-4835	221	12	,	,	PUNCT
ejpam-4835	221	13	a	a	DET
ejpam-4835	221	14	yousif	yousif	NOUN
ejpam-4835	221	15	,	,	PUNCT
ejpam-4835	221	16	and	and	CCONJ
ejpam-4835	221	17	z	z	PROPN
ejpam-4835	221	18	khan	khan	PROPN
ejpam-4835	221	19	.	.	PUNCT
ejpam-4835	222	1	mathematical	mathematical	ADJ
ejpam-4835	222	2	modeling	modeling	NOUN
ejpam-4835	222	3	and	and	CCONJ
ejpam-4835	222	4	stability	stability	NOUN
ejpam-4835	222	5	analysis	analysis	NOUN
ejpam-4835	222	6	of	of	ADP
ejpam-4835	222	7	the	the	DET
ejpam-4835	222	8	dynamics	dynamic	NOUN
ejpam-4835	222	9	of	of	ADP
ejpam-4835	222	10	monkeypox	monkeypox	NOUN
ejpam-4835	222	11	via	via	ADP
ejpam-4835	222	12	fractional	fractional	ADJ
ejpam-4835	222	13	-	-	PUNCT
ejpam-4835	222	14	calculus	calculus	NOUN
ejpam-4835	222	15	.	.	PUNCT
ejpam-4835	223	1	fractals	fractal	NOUN
ejpam-4835	223	2	,	,	PUNCT
ejpam-4835	223	3	30(10):2240266	30(10):2240266	NUM
ejpam-4835	223	4	,	,	PUNCT
ejpam-4835	223	5	2022	2022	NUM
ejpam-4835	223	6	.	.	PUNCT
ejpam-4835	224	1	[	[	X
ejpam-4835	224	2	9	9	NUM
ejpam-4835	224	3	]	]	SYM
ejpam-4835	224	4	b	b	NOUN
ejpam-4835	224	5	almutairi	almutairi	NOUN
ejpam-4835	224	6	,	,	PUNCT
ejpam-4835	224	7	i	i	PRON
ejpam-4835	224	8	ahmad	ahmad	PROPN
ejpam-4835	224	9	,	,	PUNCT
ejpam-4835	224	10	b	b	X
ejpam-4835	224	11	almohsen	almohsen	PROPN
ejpam-4835	224	12	,	,	PUNCT
ejpam-4835	224	13	h	h	PROPN
ejpam-4835	224	14	ahmad	ahmad	PROPN
ejpam-4835	224	15	,	,	PUNCT
ejpam-4835	224	16	and	and	CCONJ
ejpam-4835	224	17	du	du	PROPN
ejpam-4835	224	18	ozsahin	ozsahin	PROPN
ejpam-4835	224	19	.	.	PUNCT
ejpam-4835	225	1	numerical	numerical	ADJ
ejpam-4835	225	2	simulations	simulation	NOUN
ejpam-4835	225	3	of	of	ADP
ejpam-4835	225	4	time	time	NOUN
ejpam-4835	225	5	-	-	PUNCT
ejpam-4835	225	6	fractional	fractional	ADJ
ejpam-4835	225	7	pdes	pde	NOUN
ejpam-4835	225	8	arising	arise	VERB
ejpam-4835	225	9	in	in	ADP
ejpam-4835	225	10	mathematics	mathematic	NOUN
ejpam-4835	225	11	and	and	CCONJ
ejpam-4835	225	12	physics	physics	NOUN
ejpam-4835	225	13	using	use	VERB
ejpam-4835	225	14	the	the	DET
ejpam-4835	225	15	local	local	ADJ
ejpam-4835	225	16	meshless	meshless	ADJ
ejpam-4835	225	17	differential	differential	ADJ
ejpam-4835	225	18	quadrature	quadrature	NOUN
ejpam-4835	225	19	method	method	NOUN
ejpam-4835	225	20	.	.	PUNCT
ejpam-4835	226	1	thermal	thermal	ADJ
ejpam-4835	226	2	science	science	NOUN
ejpam-4835	226	3	,	,	PUNCT
ejpam-4835	226	4	27(spec	27(spec	NUM
ejpam-4835	226	5	.	.	PUNCT
ejpam-4835	227	1	issue	issue	NOUN
ejpam-4835	227	2	1):263–272	1):263–272	NUM
ejpam-4835	227	3	,	,	PUNCT
ejpam-4835	227	4	2023	2023	NUM
ejpam-4835	227	5	.	.	PUNCT
ejpam-4835	228	1	[	[	X
ejpam-4835	228	2	10	10	NUM
ejpam-4835	228	3	]	]	X
ejpam-4835	228	4	m	m	VERB
ejpam-4835	228	5	boulbrachene	boulbrachene	ADJ
ejpam-4835	228	6	and	and	CCONJ
ejpam-4835	228	7	m	m	VERB
ejpam-4835	228	8	haiour	haiour	NOUN
ejpam-4835	228	9	.	.	PUNCT
ejpam-4835	229	1	the	the	DET
ejpam-4835	229	2	finite	finite	PROPN
ejpam-4835	229	3	element	element	PROPN
ejpam-4835	229	4	approximation	approximation	NOUN
ejpam-4835	229	5	of	of	ADP
ejpam-4835	229	6	hamilton	hamilton	PROPN
ejpam-4835	229	7	-	-	PUNCT
ejpam-4835	229	8	jacobibellman	jacobibellman	NOUN
ejpam-4835	229	9	equations	equation	NOUN
ejpam-4835	229	10	.	.	PUNCT
ejpam-4835	230	1	computers	computer	NOUN
ejpam-4835	230	2	&	&	CCONJ
ejpam-4835	230	3	mathematics	mathematics	PROPN
ejpam-4835	230	4	with	with	ADP
ejpam-4835	230	5	applications	application	NOUN
ejpam-4835	230	6	,	,	PUNCT
ejpam-4835	230	7	41(7	41(7	NOUN
ejpam-4835	230	8	-	-	PUNCT
ejpam-4835	230	9	8):993–1007	8):993–1007	NUM
ejpam-4835	230	10	,	,	PUNCT
ejpam-4835	230	11	2001	2001	NUM
ejpam-4835	230	12	.	.	PUNCT
ejpam-4835	231	1	[	[	X
ejpam-4835	231	2	11	11	NUM
ejpam-4835	231	3	]	]	X
ejpam-4835	231	4	f	f	PROPN
ejpam-4835	231	5	brezzi	brezzi	NOUN
ejpam-4835	231	6	and	and	CCONJ
ejpam-4835	231	7	la	la	PRON
ejpam-4835	231	8	caffarelli	caffarelli	PROPN
ejpam-4835	231	9	.	.	PUNCT
ejpam-4835	232	1	convergence	convergence	NOUN
ejpam-4835	232	2	of	of	ADP
ejpam-4835	232	3	the	the	DET
ejpam-4835	232	4	discrete	discrete	ADJ
ejpam-4835	232	5	free	free	ADJ
ejpam-4835	232	6	boundaries	boundary	NOUN
ejpam-4835	232	7	for	for	ADP
ejpam-4835	232	8	finite	finite	ADJ
ejpam-4835	232	9	element	element	NOUN
ejpam-4835	232	10	approximations	approximation	NOUN
ejpam-4835	232	11	.	.	PUNCT
ejpam-4835	233	1	rairo	rairo	PROPN
ejpam-4835	233	2	.	.	PUNCT
ejpam-4835	234	1	analyse	analyse	PROPN
ejpam-4835	234	2	numérique	numérique	PROPN
ejpam-4835	234	3	,	,	PUNCT
ejpam-4835	234	4	17(4):385–395	17(4):385–395	PROPN
ejpam-4835	234	5	,	,	PUNCT
ejpam-4835	234	6	1983	1983	NUM
ejpam-4835	234	7	.	.	PUNCT
ejpam-4835	235	1	[	[	X
ejpam-4835	235	2	12	12	NUM
ejpam-4835	235	3	]	]	X
ejpam-4835	235	4	pg	pg	NOUN
ejpam-4835	235	5	ciarlet	ciarlet	VERB
ejpam-4835	235	6	and	and	CCONJ
ejpam-4835	235	7	p	p	NOUN
ejpam-4835	235	8	-	-	PUNCT
ejpam-4835	235	9	a	a	DET
ejpam-4835	235	10	raviart	raviart	NOUN
ejpam-4835	235	11	.	.	PUNCT
ejpam-4835	236	1	maximum	maximum	ADJ
ejpam-4835	236	2	principle	principle	NOUN
ejpam-4835	236	3	and	and	CCONJ
ejpam-4835	236	4	uniform	uniform	ADJ
ejpam-4835	236	5	convergence	convergence	NOUN
ejpam-4835	236	6	for	for	ADP
ejpam-4835	236	7	the	the	DET
ejpam-4835	236	8	finite	finite	ADJ
ejpam-4835	236	9	element	element	NOUN
ejpam-4835	236	10	method	method	NOUN
ejpam-4835	236	11	.	.	PUNCT
ejpam-4835	237	1	computer	computer	NOUN
ejpam-4835	237	2	methods	method	NOUN
ejpam-4835	237	3	in	in	ADP
ejpam-4835	237	4	applied	applied	ADJ
ejpam-4835	237	5	mechanics	mechanic	NOUN
ejpam-4835	237	6	and	and	CCONJ
ejpam-4835	237	7	engineering	engineering	NOUN
ejpam-4835	237	8	,	,	PUNCT
ejpam-4835	237	9	2(1):17–31	2(1):17–31	NUM
ejpam-4835	237	10	,	,	PUNCT
ejpam-4835	237	11	1973	1973	NUM
ejpam-4835	237	12	.	.	PUNCT
ejpam-4835	238	1	references	reference	NOUN
ejpam-4835	238	2	1968	1968	NUM
ejpam-4835	238	3	[	[	X
ejpam-4835	238	4	13	13	NUM
ejpam-4835	238	5	]	]	PUNCT
ejpam-4835	238	6	p.	p.	PROPN
ejpam-4835	238	7	cortey	cortey	PROPN
ejpam-4835	238	8	-	-	PUNCT
ejpam-4835	238	9	dumont	dumont	NOUN
ejpam-4835	238	10	.	.	PUNCT
ejpam-4835	239	1	on	on	ADP
ejpam-4835	239	2	the	the	DET
ejpam-4835	239	3	finite	finite	PROPN
ejpam-4835	239	4	element	element	NOUN
ejpam-4835	239	5	approximation	approximation	NOUN
ejpam-4835	239	6	in	in	ADP
ejpam-4835	239	7	the	the	DET
ejpam-4835	239	8	l∞	l∞	NOUN
ejpam-4835	239	9	norm	norm	NOUN
ejpam-4835	239	10	of	of	ADP
ejpam-4835	239	11	variational	variational	ADJ
ejpam-4835	239	12	inequalities	inequality	NOUN
ejpam-4835	239	13	with	with	ADP
ejpam-4835	239	14	nonlinear	nonlinear	ADJ
ejpam-4835	239	15	operators	operator	NOUN
ejpam-4835	239	16	.	.	PUNCT
ejpam-4835	240	1	numerische	numerische	PROPN
ejpam-4835	240	2	mathematik	mathematik	PROPN
ejpam-4835	240	3	,	,	PUNCT
ejpam-4835	240	4	47:45–57	47:45–57	PROPN
ejpam-4835	240	5	,	,	PUNCT
ejpam-4835	240	6	1985	1985	NUM
ejpam-4835	240	7	.	.	PUNCT
ejpam-4835	241	1	[	[	X
ejpam-4835	241	2	14	14	NUM
ejpam-4835	241	3	]	]	X
ejpam-4835	241	4	p	p	X
ejpam-4835	241	5	cortey	cortey	PROPN
ejpam-4835	241	6	-	-	PUNCT
ejpam-4835	241	7	dumont	dumont	NOUN
ejpam-4835	241	8	and	and	CCONJ
ejpam-4835	241	9	s	s	NOUN
ejpam-4835	241	10	i’analyse	i’analyse	PROPN
ejpam-4835	241	11	.	.	PUNCT
ejpam-4835	242	1	numérique	numérique	PROPN
ejpam-4835	242	2	des	des	X
ejpam-4835	242	3	equations	equations	PROPN
ejpam-4835	242	4	de	de	PROPN
ejpam-4835	242	5	hamiltonjacobibellman	hamiltonjacobibellman	PROPN
ejpam-4835	242	6	.	.	PUNCT
ejpam-4835	243	1	mathematical	mathematical	ADJ
ejpam-4835	243	2	methods	method	NOUN
ejpam-4835	243	3	in	in	ADP
ejpam-4835	243	4	the	the	DET
ejpam-4835	243	5	applied	apply	VERB
ejpam-4835	243	6	sciences	science	NOUN
ejpam-4835	243	7	,	,	PUNCT
ejpam-4835	243	8	9:198–209	9:198–209	NOUN
ejpam-4835	243	9	,	,	PUNCT
ejpam-4835	243	10	1987	1987	NUM
ejpam-4835	243	11	.	.	PUNCT
ejpam-4835	244	1	[	[	X
ejpam-4835	244	2	15	15	NUM
ejpam-4835	244	3	]	]	X
ejpam-4835	244	4	w	w	PROPN
ejpam-4835	244	5	hackbusch	hackbusch	PROPN
ejpam-4835	244	6	.	.	PUNCT
ejpam-4835	245	1	multi	multi	ADJ
ejpam-4835	245	2	-	-	ADJ
ejpam-4835	245	3	grid	grid	ADJ
ejpam-4835	245	4	methods	method	NOUN
ejpam-4835	245	5	and	and	CCONJ
ejpam-4835	245	6	applications	application	NOUN
ejpam-4835	245	7	.	.	PUNCT
ejpam-4835	246	1	,	,	PUNCT
ejpam-4835	246	2	volume	volume	NOUN
ejpam-4835	246	3	4	4	NUM
ejpam-4835	246	4	.	.	PUNCT
ejpam-4835	246	5	springer	springer	NOUN
ejpam-4835	246	6	science	science	PROPN
ejpam-4835	246	7	&	&	CCONJ
ejpam-4835	246	8	business	business	NOUN
ejpam-4835	246	9	media	medium	NOUN
ejpam-4835	246	10	,	,	PUNCT
ejpam-4835	246	11	2013	2013	NUM
ejpam-4835	246	12	.	.	PUNCT
ejpam-4835	247	1	[	[	X
ejpam-4835	247	2	16	16	NUM
ejpam-4835	247	3	]	]	X
ejpam-4835	247	4	w	w	NOUN
ejpam-4835	247	5	hackbusch	hackbusch	PROPN
ejpam-4835	247	6	and	and	CCONJ
ejpam-4835	247	7	hd	hd	PROPN
ejpam-4835	247	8	mittelmann	mittelmann	NOUN
ejpam-4835	247	9	.	.	PUNCT
ejpam-4835	248	1	on	on	ADP
ejpam-4835	248	2	multi	multi	ADJ
ejpam-4835	248	3	-	-	ADJ
ejpam-4835	248	4	grid	grid	ADJ
ejpam-4835	248	5	methods	method	NOUN
ejpam-4835	248	6	for	for	ADP
ejpam-4835	248	7	variational	variational	ADJ
ejpam-4835	248	8	inequalities	inequality	NOUN
ejpam-4835	248	9	.	.	PUNCT
ejpam-4835	249	1	numerische	numerische	PROPN
ejpam-4835	249	2	mathematik	mathematik	PROPN
ejpam-4835	249	3	,	,	PUNCT
ejpam-4835	249	4	42:65–76	42:65–76	PROPN
ejpam-4835	249	5	,	,	PUNCT
ejpam-4835	249	6	1983	1983	NUM
ejpam-4835	249	7	.	.	PUNCT
ejpam-4835	250	1	[	[	X
ejpam-4835	250	2	17	17	NUM
ejpam-4835	250	3	]	]	X
ejpam-4835	250	4	m	m	VERB
ejpam-4835	250	5	haiour	haiour	NOUN
ejpam-4835	250	6	.	.	PUNCT
ejpam-4835	250	7	etude	etude	PROPN
ejpam-4835	250	8	de	de	X
ejpam-4835	250	9	la	la	PROPN
ejpam-4835	250	10	convergence	convergence	NOUN
ejpam-4835	250	11	uniforme	uniforme	X
ejpam-4835	250	12	de	de	X
ejpam-4835	250	13	la	la	PROPN
ejpam-4835	250	14	méthode	méthode	PROPN
ejpam-4835	250	15	multigrilles	multigrilles	PROPN
ejpam-4835	250	16	appliquées	appliquée	NOUN
ejpam-4835	250	17	aux	aux	X
ejpam-4835	250	18	problèmes	problème	VERB
ejpam-4835	250	19	frontières	frontières	ADP
ejpam-4835	250	20	libres	libre	NOUN
ejpam-4835	250	21	.	.	PUNCT
ejpam-4835	251	1	phd	phd	NOUN
ejpam-4835	251	2	thesis	thesis	NOUN
ejpam-4835	251	3	,	,	PUNCT
ejpam-4835	251	4	université	université	ADJ
ejpam-4835	251	5	de	de	X
ejpam-4835	251	6	annaba	annaba	PROPN
ejpam-4835	251	7	-	-	PUNCT
ejpam-4835	251	8	badji	badji	PROPN
ejpam-4835	251	9	mokhtar	mokhtar	PROPN
ejpam-4835	251	10	,	,	PUNCT
ejpam-4835	251	11	2004	2004	NUM
ejpam-4835	251	12	.	.	PUNCT
ejpam-4835	252	1	[	[	X
ejpam-4835	252	2	18	18	NUM
ejpam-4835	252	3	]	]	X
ejpam-4835	252	4	r	r	NOUN
ejpam-4835	252	5	hw	hw	PROPN
ejpam-4835	252	6	hoppe	hoppe	PROPN
ejpam-4835	252	7	.	.	PUNCT
ejpam-4835	253	1	multi	multi	ADJ
ejpam-4835	253	2	-	-	ADJ
ejpam-4835	253	3	grid	grid	ADJ
ejpam-4835	253	4	methods	method	NOUN
ejpam-4835	253	5	for	for	ADP
ejpam-4835	253	6	hamilton	hamilton	PROPN
ejpam-4835	253	7	-	-	PUNCT
ejpam-4835	253	8	jacobi	jacobi	PROPN
ejpam-4835	253	9	-	-	PUNCT
ejpam-4835	253	10	bellman	bellman	PROPN
ejpam-4835	253	11	equations	equation	NOUN
ejpam-4835	253	12	.	.	PUNCT
ejpam-4835	254	1	numerische	numerische	PROPN
ejpam-4835	254	2	mathematik	mathematik	PROPN
ejpam-4835	254	3	,	,	PUNCT
ejpam-4835	254	4	49:239–254	49:239–254	PROPN
ejpam-4835	254	5	,	,	PUNCT
ejpam-4835	254	6	1986	1986	NUM
ejpam-4835	254	7	.	.	PUNCT
ejpam-4835	255	1	[	[	X
ejpam-4835	255	2	19	19	NUM
ejpam-4835	255	3	]	]	X
ejpam-4835	255	4	r	r	NOUN
ejpam-4835	255	5	hw	hw	PROPN
ejpam-4835	255	6	hoppe	hoppe	PROPN
ejpam-4835	255	7	.	.	PUNCT
ejpam-4835	256	1	multigrid	multigrid	ADJ
ejpam-4835	256	2	algorithms	algorithm	NOUN
ejpam-4835	256	3	for	for	ADP
ejpam-4835	256	4	variational	variational	ADJ
ejpam-4835	256	5	inequalities	inequality	NOUN
ejpam-4835	256	6	.	.	PUNCT
ejpam-4835	257	1	siam	siam	PROPN
ejpam-4835	257	2	journal	journal	PROPN
ejpam-4835	257	3	on	on	ADP
ejpam-4835	257	4	numerical	numerical	ADJ
ejpam-4835	257	5	analysis	analysis	NOUN
ejpam-4835	257	6	,	,	PUNCT
ejpam-4835	257	7	24(5):1046–1065	24(5):1046–1065	NUM
ejpam-4835	257	8	,	,	PUNCT
ejpam-4835	257	9	1987	1987	NUM
ejpam-4835	257	10	.	.	PUNCT
ejpam-4835	258	1	[	[	X
ejpam-4835	258	2	20	20	NUM
ejpam-4835	258	3	]	]	X
ejpam-4835	258	4	h	h	PROPN
ejpam-4835	258	5	irshad	irshad	VERB
ejpam-4835	258	6	,	,	PUNCT
ejpam-4835	258	7	m	m	PROPN
ejpam-4835	258	8	shakeel	shakeel	PROPN
ejpam-4835	258	9	,	,	PUNCT
ejpam-4835	258	10	i	i	PRON
ejpam-4835	258	11	ahmad	ahmad	PROPN
ejpam-4835	258	12	,	,	PUNCT
ejpam-4835	258	13	h	h	PROPN
ejpam-4835	258	14	ahmad	ahmad	PROPN
ejpam-4835	258	15	,	,	PUNCT
ejpam-4835	258	16	c	c	PROPN
ejpam-4835	258	17	tearnbucha	tearnbucha	NOUN
ejpam-4835	258	18	,	,	PUNCT
ejpam-4835	258	19	and	and	CCONJ
ejpam-4835	258	20	w	w	NOUN
ejpam-4835	258	21	sudsutad	sudsutad	NOUN
ejpam-4835	258	22	.	.	PUNCT
ejpam-4835	259	1	simulation	simulation	NOUN
ejpam-4835	259	2	of	of	ADP
ejpam-4835	259	3	generalized	generalized	ADJ
ejpam-4835	259	4	time	time	NOUN
ejpam-4835	259	5	fractional	fractional	PROPN
ejpam-4835	259	6	gardner	gardner	NOUN
ejpam-4835	259	7	equation	equation	NOUN
ejpam-4835	259	8	utilizing	utilize	VERB
ejpam-4835	259	9	in	in	ADP
ejpam-4835	259	10	plasma	plasma	NOUN
ejpam-4835	259	11	physics	physics	NOUN
ejpam-4835	259	12	for	for	ADP
ejpam-4835	259	13	non	non	ADJ
ejpam-4835	259	14	-	-	ADJ
ejpam-4835	259	15	linear	linear	ADJ
ejpam-4835	259	16	propagation	propagation	NOUN
ejpam-4835	259	17	of	of	ADP
ejpam-4835	259	18	ion	ion	NOUN
ejpam-4835	259	19	-	-	PUNCT
ejpam-4835	259	20	acoustic	acoustic	ADJ
ejpam-4835	259	21	waves	wave	NOUN
ejpam-4835	259	22	.	.	PUNCT
ejpam-4835	260	1	thermal	thermal	ADJ
ejpam-4835	260	2	science	science	NOUN
ejpam-4835	260	3	,	,	PUNCT
ejpam-4835	260	4	27(spec	27(spec	NUM
ejpam-4835	260	5	.	.	PUNCT
ejpam-4835	260	6	issue	issue	NOUN
ejpam-4835	260	7	1):121–128	1):121–128	NUM
ejpam-4835	260	8	,	,	PUNCT
ejpam-4835	260	9	2023	2023	NUM
ejpam-4835	260	10	.	.	PUNCT
ejpam-4835	261	1	[	[	X
ejpam-4835	261	2	21	21	NUM
ejpam-4835	261	3	]	]	X
ejpam-4835	261	4	r	r	X
ejpam-4835	261	5	jan	jan	PROPN
ejpam-4835	261	6	,	,	PUNCT
ejpam-4835	261	7	a	a	DET
ejpam-4835	261	8	khan	khan	PROPN
ejpam-4835	261	9	,	,	PUNCT
ejpam-4835	261	10	s	s	PART
ejpam-4835	261	11	boulaaras	boulaaras	NOUN
ejpam-4835	261	12	,	,	PUNCT
ejpam-4835	261	13	and	and	CCONJ
ejpam-4835	261	14	sa	sa	PROPN
ejpam-4835	261	15	zubair	zubair	PROPN
ejpam-4835	261	16	.	.	PUNCT
ejpam-4835	262	1	dynamical	dynamical	ADJ
ejpam-4835	262	2	behaviour	behaviour	NOUN
ejpam-4835	262	3	and	and	CCONJ
ejpam-4835	262	4	chaotic	chaotic	ADJ
ejpam-4835	262	5	phenomena	phenomenon	NOUN
ejpam-4835	262	6	of	of	ADP
ejpam-4835	262	7	hiv	hiv	PROPN
ejpam-4835	262	8	infection	infection	NOUN
ejpam-4835	262	9	through	through	ADP
ejpam-4835	262	10	fractional	fractional	ADJ
ejpam-4835	262	11	calculus	calculus	NOUN
ejpam-4835	262	12	.	.	PUNCT
ejpam-4835	263	1	discrete	discrete	ADJ
ejpam-4835	263	2	dynamics	dynamic	NOUN
ejpam-4835	263	3	in	in	ADP
ejpam-4835	263	4	nature	nature	NOUN
ejpam-4835	263	5	and	and	CCONJ
ejpam-4835	263	6	society	society	NOUN
ejpam-4835	263	7	,	,	PUNCT
ejpam-4835	263	8	2022	2022	NUM
ejpam-4835	263	9	,	,	PUNCT
ejpam-4835	263	10	2022	2022	NUM
ejpam-4835	263	11	.	.	PUNCT
ejpam-4835	264	1	[	[	X
ejpam-4835	264	2	22	22	NUM
ejpam-4835	264	3	]	]	X
ejpam-4835	264	4	d	d	X
ejpam-4835	264	5	kinderlehrer	kinderlehrer	NOUN
ejpam-4835	264	6	and	and	CCONJ
ejpam-4835	264	7	g	g	PROPN
ejpam-4835	264	8	stampacchia	stampacchia	X
ejpam-4835	264	9	.	.	PUNCT
ejpam-4835	265	1	an	an	DET
ejpam-4835	265	2	introduction	introduction	NOUN
ejpam-4835	265	3	to	to	ADP
ejpam-4835	265	4	variational	variational	ADJ
ejpam-4835	265	5	inequalities	inequality	NOUN
ejpam-4835	265	6	and	and	CCONJ
ejpam-4835	265	7	their	their	PRON
ejpam-4835	265	8	applications	application	NOUN
ejpam-4835	265	9	.	.	PUNCT
ejpam-4835	266	1	siam	siam	PROPN
ejpam-4835	266	2	,	,	PUNCT
ejpam-4835	266	3	2000	2000	NUM
ejpam-4835	266	4	.	.	PUNCT
ejpam-4835	267	1	[	[	X
ejpam-4835	267	2	23	23	NUM
ejpam-4835	267	3	]	]	X
ejpam-4835	267	4	j	j	PROPN
ejpam-4835	267	5	-	-	PROPN
ejpam-4835	267	6	f	f	PROPN
ejpam-4835	267	7	li	li	PROPN
ejpam-4835	267	8	,	,	PUNCT
ejpam-4835	267	9	i	i	PRON
ejpam-4835	267	10	ahmad	ahmad	PROPN
ejpam-4835	267	11	,	,	PUNCT
ejpam-4835	267	12	h	h	PROPN
ejpam-4835	267	13	ahmad	ahmad	PROPN
ejpam-4835	267	14	,	,	PUNCT
ejpam-4835	267	15	dshah	dshah	PROPN
ejpam-4835	267	16	,	,	PUNCT
ejpam-4835	267	17	y	y	PROPN
ejpam-4835	267	18	-	-	PUNCT
ejpam-4835	267	19	m	m	PROPN
ejpam-4835	267	20	chu	chu	PROPN
ejpam-4835	267	21	,	,	PUNCT
ejpam-4835	267	22	p	p	NOUN
ejpam-4835	267	23	thounthong	thounthong	NOUN
ejpam-4835	267	24	,	,	PUNCT
ejpam-4835	267	25	and	and	CCONJ
ejpam-4835	267	26	m	m	PROPN
ejpam-4835	267	27	ayaz	ayaz	PROPN
ejpam-4835	267	28	.	.	PUNCT
ejpam-4835	268	1	numerical	numerical	ADJ
ejpam-4835	268	2	solution	solution	NOUN
ejpam-4835	268	3	of	of	ADP
ejpam-4835	268	4	two	two	NUM
ejpam-4835	268	5	-	-	PUNCT
ejpam-4835	268	6	term	term	NOUN
ejpam-4835	268	7	time	time	NOUN
ejpam-4835	268	8	-	-	PUNCT
ejpam-4835	268	9	fractional	fractional	ADJ
ejpam-4835	268	10	pde	pde	NOUN
ejpam-4835	268	11	models	model	NOUN
ejpam-4835	268	12	arising	arise	VERB
ejpam-4835	268	13	in	in	ADP
ejpam-4835	268	14	mathematical	mathematical	ADJ
ejpam-4835	268	15	physics	physics	NOUN
ejpam-4835	268	16	using	use	VERB
ejpam-4835	268	17	local	local	ADJ
ejpam-4835	268	18	meshless	meshless	ADJ
ejpam-4835	268	19	method	method	NOUN
ejpam-4835	268	20	.	.	PUNCT
ejpam-4835	269	1	open	open	ADJ
ejpam-4835	269	2	physics	physics	PROPN
ejpam-4835	269	3	,	,	PUNCT
ejpam-4835	269	4	18(1):1063–1072	18(1):1063–1072	NUM
ejpam-4835	269	5	,	,	PUNCT
ejpam-4835	269	6	2020	2020	NUM
ejpam-4835	269	7	.	.	PUNCT
ejpam-4835	270	1	[	[	X
ejpam-4835	270	2	24	24	NUM
ejpam-4835	270	3	]	]	X
ejpam-4835	270	4	h	h	PROPN
ejpam-4835	270	5	ahmad	ahmad	PROPN
ejpam-4835	270	6	m	m	VERB
ejpam-4835	270	7	adel	adel	PROPN
ejpam-4835	270	8	,	,	PUNCT
ejpam-4835	270	9	me	i	PRON
ejpam-4835	270	10	ramadan	ramadan	PROPN
ejpam-4835	270	11	and	and	CCONJ
ejpam-4835	270	12	t	t	PROPN
ejpam-4835	270	13	botmart	botmart	NOUN
ejpam-4835	270	14	.	.	PUNCT
ejpam-4835	271	1	sobolev	sobolev	NOUN
ejpam-4835	271	2	-	-	PUNCT
ejpam-4835	271	3	type	type	NOUN
ejpam-4835	271	4	nonlinear	nonlinear	ADJ
ejpam-4835	271	5	hilfer	hilfer	NOUN
ejpam-4835	271	6	fractional	fractional	ADJ
ejpam-4835	271	7	stochastic	stochastic	ADJ
ejpam-4835	271	8	differential	differential	ADJ
ejpam-4835	271	9	equations	equation	NOUN
ejpam-4835	271	10	with	with	ADP
ejpam-4835	271	11	noninstantaneous	noninstantaneous	ADJ
ejpam-4835	271	12	impulsive	impulsive	NOUN
ejpam-4835	271	13	.	.	PUNCT
ejpam-4835	272	1	aims	aim	VERB
ejpam-4835	272	2	mathematics	mathematic	NOUN
ejpam-4835	272	3	,	,	PUNCT
ejpam-4835	272	4	7(11):20105–20125	7(11):20105–20125	NUM
ejpam-4835	272	5	,	,	PUNCT
ejpam-4835	272	6	2022	2022	NUM
ejpam-4835	272	7	.	.	PUNCT
ejpam-4835	273	1	[	[	X
ejpam-4835	273	2	25	25	NUM
ejpam-4835	273	3	]	]	PUNCT
ejpam-4835	273	4	a.	a.	NOUN
ejpam-4835	273	5	reusken	reusken	PROPN
ejpam-4835	273	6	.	.	PUNCT
ejpam-4835	274	1	on	on	ADP
ejpam-4835	274	2	maximum	maximum	ADJ
ejpam-4835	274	3	norm	norm	NOUN
ejpam-4835	274	4	convergcnce	convergcnce	NOUN
ejpam-4835	274	5	of	of	ADP
ejpam-4835	274	6	multigrids	multigrid	NOUN
ejpam-4835	274	7	methods	method	NOUN
ejpam-4835	274	8	for	for	ADP
ejpam-4835	274	9	elliptic	elliptic	ADJ
ejpam-4835	274	10	boundary	boundary	ADJ
ejpam-4835	274	11	value	value	NOUN
ejpam-4835	274	12	problems	problem	NOUN
ejpam-4835	274	13	.	.	PUNCT
ejpam-4835	275	1	siam	siam	PROPN
ejpam-4835	275	2	journal	journal	PROPN
ejpam-4835	275	3	on	on	ADP
ejpam-4835	275	4	numerical	numerical	ADJ
ejpam-4835	275	5	analysis	analysis	NOUN
ejpam-4835	275	6	,	,	PUNCT
ejpam-4835	275	7	29(6):1569–1578	29(6):1569–1578	NUM
ejpam-4835	275	8	,	,	PUNCT
ejpam-4835	275	9	1992	1992	NUM
ejpam-4835	275	10	.	.	PUNCT
ejpam-4835	276	1	[	[	X
ejpam-4835	276	2	26	26	NUM
ejpam-4835	276	3	]	]	PUNCT
ejpam-4835	276	4	a	a	DET
ejpam-4835	276	5	reusken	reusken	NOUN
ejpam-4835	276	6	.	.	PUNCT
ejpam-4835	276	7	introduction	introduction	NOUN
ejpam-4835	276	8	to	to	ADP
ejpam-4835	276	9	multigrid	multigrid	ADJ
ejpam-4835	276	10	methods	method	NOUN
ejpam-4835	276	11	for	for	ADP
ejpam-4835	276	12	elliptic	elliptic	ADJ
ejpam-4835	276	13	boundary	boundary	ADJ
ejpam-4835	276	14	value	value	NOUN
ejpam-4835	276	15	problems	problem	NOUN
ejpam-4835	276	16	.	.	PUNCT
ejpam-4835	277	1	inst	inst	NOUN
ejpam-4835	277	2	.	.	PUNCT
ejpam-4835	278	1	für	für	PROPN
ejpam-4835	278	2	geometrie	geometrie	PROPN
ejpam-4835	278	3	und	und	PROPN
ejpam-4835	278	4	praktische	praktische	PROPN
ejpam-4835	278	5	mathematik	mathematik	PROPN
ejpam-4835	278	6	,	,	PUNCT
ejpam-4835	278	7	2008	2008	NUM
ejpam-4835	278	8	.	.	PUNCT
ejpam-4835	279	1	references	reference	NOUN
ejpam-4835	279	2	1969	1969	NUM
ejpam-4835	280	1	[	[	X
ejpam-4835	280	2	27	27	NUM
ejpam-4835	280	3	]	]	PUNCT
ejpam-4835	280	4	na	na	ADP
ejpam-4835	280	5	shah	shah	PROPN
ejpam-4835	280	6	,	,	PUNCT
ejpam-4835	280	7	i	i	PRON
ejpam-4835	280	8	ahmad	ahmad	PROPN
ejpam-4835	280	9	,	,	PUNCT
ejpam-4835	280	10	o	o	PROPN
ejpam-4835	280	11	bazighifan	bazighifan	NOUN
ejpam-4835	280	12	,	,	PUNCT
ejpam-4835	280	13	ae	ae	PROPN
ejpam-4835	280	14	abouelregal	abouelregal	ADJ
ejpam-4835	280	15	,	,	PUNCT
ejpam-4835	280	16	and	and	CCONJ
ejpam-4835	280	17	h	h	PROPN
ejpam-4835	280	18	ahmad	ahmad	PROPN
ejpam-4835	280	19	.	.	PUNCT
ejpam-4835	281	1	multistage	multistage	NOUN
ejpam-4835	281	2	optimal	optimal	ADJ
ejpam-4835	281	3	homotopy	homotopy	NOUN
ejpam-4835	281	4	asymptotic	asymptotic	ADJ
ejpam-4835	281	5	method	method	NOUN
ejpam-4835	281	6	for	for	ADP
ejpam-4835	281	7	the	the	DET
ejpam-4835	281	8	nonlinear	nonlinear	ADJ
ejpam-4835	281	9	riccati	riccati	PROPN
ejpam-4835	281	10	ordinary	ordinary	ADJ
ejpam-4835	281	11	differential	differential	ADJ
ejpam-4835	281	12	equation	equation	NOUN
ejpam-4835	281	13	in	in	ADP
ejpam-4835	281	14	nonlinear	nonlinear	ADJ
ejpam-4835	281	15	physics	physics	PROPN
ejpam-4835	281	16	.	.	PUNCT
ejpam-4835	282	1	applied	apply	VERB
ejpam-4835	282	2	mathematics	mathematic	NOUN
ejpam-4835	282	3	,	,	PUNCT
ejpam-4835	282	4	14(6):1009–1016	14(6):1009–1016	NUM
ejpam-4835	282	5	,	,	PUNCT
ejpam-4835	282	6	2020	2020	NUM
ejpam-4835	282	7	.	.	PUNCT
ejpam-4835	283	1	[	[	X
ejpam-4835	283	2	28	28	NUM
ejpam-4835	283	3	]	]	X
ejpam-4835	283	4	m	m	PROPN
ejpam-4835	283	5	shakeel	shakeel	PROPN
ejpam-4835	283	6	,	,	PUNCT
ejpam-4835	283	7	i	i	PRON
ejpam-4835	283	8	hussain	hussain	VERB
ejpam-4835	283	9	,	,	PUNCT
ejpam-4835	283	10	h	h	PROPN
ejpam-4835	283	11	ahmad	ahmad	PROPN
ejpam-4835	283	12	,	,	PUNCT
ejpam-4835	283	13	i	i	PRON
ejpam-4835	283	14	ahmad	ahmad	PROPN
ejpam-4835	283	15	,	,	PUNCT
ejpam-4835	283	16	p	p	NOUN
ejpam-4835	283	17	thounthong	thounthong	NOUN
ejpam-4835	283	18	,	,	PUNCT
ejpam-4835	283	19	y	y	PROPN
ejpam-4835	283	20	-	-	PUNCT
ejpam-4835	283	21	f	f	PROPN
ejpam-4835	283	22	zhang	zhang	PROPN
ejpam-4835	283	23	,	,	PUNCT
ejpam-4835	283	24	and	and	CCONJ
ejpam-4835	283	25	yingfang	yingfang	PROPN
ejpam-4835	283	26	.	.	PUNCT
ejpam-4835	284	1	meshless	meshless	ADJ
ejpam-4835	284	2	technique	technique	NOUN
ejpam-4835	284	3	for	for	ADP
ejpam-4835	284	4	the	the	DET
ejpam-4835	284	5	solution	solution	NOUN
ejpam-4835	284	6	of	of	ADP
ejpam-4835	284	7	time	time	NOUN
ejpam-4835	284	8	-	-	PUNCT
ejpam-4835	284	9	fractional	fractional	ADJ
ejpam-4835	284	10	partial	partial	ADJ
ejpam-4835	284	11	differential	differential	NOUN
ejpam-4835	284	12	equations	equation	NOUN
ejpam-4835	284	13	having	have	VERB
ejpam-4835	284	14	real	real	ADJ
ejpam-4835	284	15	-	-	PUNCT
ejpam-4835	284	16	world	world	NOUN
ejpam-4835	284	17	applications	application	NOUN
ejpam-4835	284	18	.	.	PUNCT
ejpam-4835	285	1	journal	journal	NOUN
ejpam-4835	285	2	of	of	ADP
ejpam-4835	285	3	function	function	NOUN
ejpam-4835	285	4	spaces	space	NOUN
ejpam-4835	285	5	,	,	PUNCT
ejpam-4835	285	6	2020:1–17	2020:1–17	NUM
ejpam-4835	285	7	,	,	PUNCT
ejpam-4835	285	8	2020	2020	NUM
ejpam-4835	285	9	.	.	PUNCT
ejpam-4835	286	1	[	[	X
ejpam-4835	286	2	29	29	NUM
ejpam-4835	286	3	]	]	X
ejpam-4835	286	4	m	m	PROPN
ejpam-4835	286	5	shakeel	shakeel	PROPN
ejpam-4835	286	6	,	,	PUNCT
ejpam-4835	286	7	mn	mn	PROPN
ejpam-4835	286	8	khan	khan	PROPN
ejpam-4835	286	9	,	,	PUNCT
ejpam-4835	286	10	i	i	PROPN
ejpam-4835	286	11	ahmad	ahmad	PROPN
ejpam-4835	286	12	,	,	PUNCT
ejpam-4835	286	13	h	h	PROPN
ejpam-4835	286	14	ahmad	ahmad	PROPN
ejpam-4835	286	15	,	,	PUNCT
ejpam-4835	286	16	n	n	PRON
ejpam-4835	286	17	jarasthitikulchai	jarasthitikulchai	ADJ
ejpam-4835	286	18	,	,	PUNCT
ejpam-4835	286	19	and	and	CCONJ
ejpam-4835	286	20	w	w	NOUN
ejpam-4835	286	21	sudsutad	sudsutad	NOUN
ejpam-4835	286	22	.	.	PUNCT
ejpam-4835	287	1	local	local	ADJ
ejpam-4835	287	2	meshless	meshless	ADJ
ejpam-4835	287	3	collocation	collocation	NOUN
ejpam-4835	287	4	scheme	scheme	NOUN
ejpam-4835	287	5	for	for	ADP
ejpam-4835	287	6	numerical	numerical	ADJ
ejpam-4835	287	7	simulation	simulation	NOUN
ejpam-4835	287	8	of	of	ADP
ejpam-4835	287	9	space	space	NOUN
ejpam-4835	287	10	fractional	fractional	ADJ
ejpam-4835	287	11	pde	pde	NOUN
ejpam-4835	287	12	.	.	PUNCT
ejpam-4835	288	1	thermal	thermal	ADJ
ejpam-4835	288	2	science	science	NOUN
ejpam-4835	288	3	,	,	PUNCT
ejpam-4835	288	4	27(spec	27(spec	NUM
ejpam-4835	288	5	.	.	PUNCT
ejpam-4835	288	6	issue	issue	NOUN
ejpam-4835	288	7	1):101–109	1):101–109	NUM
ejpam-4835	288	8	,	,	PUNCT
ejpam-4835	288	9	2023	2023	NUM
ejpam-4835	288	10	.	.	PUNCT
ejpam-4835	289	1	[	[	X
ejpam-4835	289	2	30	30	NUM
ejpam-4835	289	3	]	]	PUNCT
ejpam-4835	289	4	ta	ta	ADP
ejpam-4835	289	5	sulaiman	sulaiman	PROPN
ejpam-4835	289	6	,	,	PUNCT
ejpam-4835	289	7	a	a	DET
ejpam-4835	289	8	yusuf	yusuf	PROPN
ejpam-4835	289	9	,	,	PUNCT
ejpam-4835	289	10	s	s	PART
ejpam-4835	289	11	abdel	abdel	NOUN
ejpam-4835	289	12	-	-	PUNCT
ejpam-4835	289	13	khalek	khalek	PROPN
ejpam-4835	289	14	,	,	PUNCT
ejpam-4835	289	15	m	m	PROPN
ejpam-4835	289	16	bayram	bayram	PROPN
ejpam-4835	289	17	,	,	PUNCT
ejpam-4835	289	18	and	and	CCONJ
ejpam-4835	289	19	h	h	PROPN
ejpam-4835	289	20	ahmad	ahmad	PROPN
ejpam-4835	289	21	.	.	PUNCT
ejpam-4835	290	1	nonautonomous	nonautonomous	ADJ
ejpam-4835	290	2	complex	complex	ADJ
ejpam-4835	290	3	wave	wave	NOUN
ejpam-4835	290	4	solutions	solution	NOUN
ejpam-4835	290	5	to	to	ADP
ejpam-4835	290	6	the	the	DET
ejpam-4835	290	7	(	(	PUNCT
ejpam-4835	290	8	2	2	NUM
ejpam-4835	290	9	+	+	NUM
ejpam-4835	290	10	1)-dimensional	1)-dimensional	ADJ
ejpam-4835	290	11	variable	variable	ADJ
ejpam-4835	290	12	-	-	PUNCT
ejpam-4835	290	13	coefficients	coefficient	NOUN
ejpam-4835	290	14	nonlinear	nonlinear	ADJ
ejpam-4835	290	15	chiral	chiral	ADJ
ejpam-4835	290	16	schrödinger	schrödinger	NOUN
ejpam-4835	290	17	equation	equation	NOUN
ejpam-4835	290	18	.	.	PUNCT
ejpam-4835	291	1	results	result	NOUN
ejpam-4835	291	2	in	in	ADP
ejpam-4835	291	3	physics	physics	NOUN
ejpam-4835	291	4	,	,	PUNCT
ejpam-4835	291	5	19:103604	19:103604	NUM
ejpam-4835	291	6	,	,	PUNCT
ejpam-4835	291	7	2020	2020	NUM
ejpam-4835	291	8	.	.	PUNCT
ejpam-4835	292	1	[	[	X
ejpam-4835	292	2	31	31	NUM
ejpam-4835	292	3	]	]	X
ejpam-4835	292	4	r	r	NOUN
ejpam-4835	292	5	jan	jan	PROPN
ejpam-4835	292	6	t	t	PROPN
ejpam-4835	292	7	-	-	PUNCT
ejpam-4835	292	8	q	q	NOUN
ejpam-4835	292	9	tangz	tangz	NOUN
ejpam-4835	292	10	shah	shah	NOUN
ejpam-4835	292	11	and	and	CCONJ
ejpam-4835	292	12	e	e	NOUN
ejpam-4835	292	13	alzahrani	alzahrani	NOUN
ejpam-4835	292	14	.	.	PUNCT
ejpam-4835	293	1	modeling	model	VERB
ejpam-4835	293	2	the	the	DET
ejpam-4835	293	3	dynamics	dynamic	NOUN
ejpam-4835	293	4	of	of	ADP
ejpam-4835	293	5	tumor	tumor	NOUN
ejpam-4835	293	6	–	–	PUNCT
ejpam-4835	293	7	immune	immune	ADJ
ejpam-4835	293	8	cells	cell	NOUN
ejpam-4835	293	9	interactions	interaction	NOUN
ejpam-4835	293	10	via	via	ADP
ejpam-4835	293	11	fractional	fractional	ADJ
ejpam-4835	293	12	calculus	calculus	NOUN
ejpam-4835	293	13	.	.	PUNCT
ejpam-4835	294	1	the	the	DET
ejpam-4835	294	2	european	european	PROPN
ejpam-4835	294	3	physical	physical	PROPN
ejpam-4835	294	4	journal	journal	PROPN
ejpam-4835	294	5	plus	plus	CCONJ
ejpam-4835	294	6	,	,	PUNCT
ejpam-4835	294	7	137(3):367	137(3):367	NUM
ejpam-4835	294	8	,	,	PUNCT
ejpam-4835	294	9	2022	2022	NUM
ejpam-4835	294	10	.	.	PUNCT
ejpam-4835	295	1	[	[	X
ejpam-4835	295	2	32	32	NUM
ejpam-4835	295	3	]	]	SYM
ejpam-4835	295	4	t	t	PROPN
ejpam-4835	295	5	-	-	PUNCT
ejpam-4835	295	6	q	q	NOUN
ejpam-4835	295	7	tang	tang	NOUN
ejpam-4835	295	8	,	,	PUNCT
ejpam-4835	295	9	z	z	NOUN
ejpam-4835	295	10	shah	shah	NOUN
ejpam-4835	295	11	,	,	PUNCT
ejpam-4835	295	12	r	r	PROPN
ejpam-4835	295	13	jan	jan	PROPN
ejpam-4835	295	14	,	,	PUNCT
ejpam-4835	295	15	w	w	NOUN
ejpam-4835	295	16	deebani	deebani	PROPN
ejpam-4835	295	17	,	,	PUNCT
ejpam-4835	295	18	and	and	CCONJ
ejpam-4835	295	19	m	m	PROPN
ejpam-4835	295	20	shutaywi	shutaywi	NOUN
ejpam-4835	295	21	meshal	meshal	NOUN
ejpam-4835	295	22	.	.	PUNCT
ejpam-4835	296	1	a	a	DET
ejpam-4835	296	2	robust	robust	ADJ
ejpam-4835	296	3	study	study	NOUN
ejpam-4835	296	4	to	to	PART
ejpam-4835	296	5	conceptualize	conceptualize	VERB
ejpam-4835	296	6	the	the	DET
ejpam-4835	296	7	interactions	interaction	NOUN
ejpam-4835	296	8	of	of	ADP
ejpam-4835	296	9	cd4	cd4	PROPN
ejpam-4835	296	10	+	+	PROPN
ejpam-4835	296	11	t	t	PROPN
ejpam-4835	296	12	-	-	PUNCT
ejpam-4835	296	13	cells	cell	NOUN
ejpam-4835	296	14	and	and	CCONJ
ejpam-4835	296	15	human	human	ADJ
ejpam-4835	296	16	immunodeficiency	immunodeficiency	NOUN
ejpam-4835	296	17	virus	virus	NOUN
ejpam-4835	296	18	via	via	ADP
ejpam-4835	296	19	fractional	fractional	ADJ
ejpam-4835	296	20	-	-	PUNCT
ejpam-4835	296	21	calculus	calculus	NOUN
ejpam-4835	296	22	.	.	PUNCT
ejpam-4835	297	1	physica	physica	PROPN
ejpam-4835	297	2	scripta	scripta	PROPN
ejpam-4835	297	3	,	,	PUNCT
ejpam-4835	297	4	96(12):125231	96(12):125231	NUM
ejpam-4835	297	5	,	,	PUNCT
ejpam-4835	297	6	2021	2021	NUM
ejpam-4835	297	7	.	.	PUNCT
ejpam-4835	298	1	[	[	X
ejpam-4835	298	2	33	33	NUM
ejpam-4835	298	3	]	]	X
ejpam-4835	298	4	f	f	PROPN
ejpam-4835	298	5	wang	wang	PROPN
ejpam-4835	298	6	,	,	PUNCT
ejpam-4835	298	7	i	i	PROPN
ejpam-4835	298	8	ahmad	ahmad	PROPN
ejpam-4835	298	9	,	,	PUNCT
ejpam-4835	298	10	h	h	PROPN
ejpam-4835	298	11	ahmad	ahmad	PROPN
ejpam-4835	298	12	,	,	PUNCT
ejpam-4835	298	13	md	md	PROPN
ejpam-4835	298	14	alsulami	alsulami	NOUN
ejpam-4835	298	15	,	,	PUNCT
ejpam-4835	298	16	ks	ks	PROPN
ejpam-4835	298	17	alimgeer	alimgeer	PROPN
ejpam-4835	298	18	,	,	PUNCT
ejpam-4835	298	19	c	c	PROPN
ejpam-4835	298	20	cesarano	cesarano	ADJ
ejpam-4835	298	21	,	,	PUNCT
ejpam-4835	298	22	and	and	CCONJ
ejpam-4835	298	23	ta	ta	AUX
ejpam-4835	298	24	nofal	nofal	NOUN
ejpam-4835	298	25	.	.	PUNCT
ejpam-4835	299	1	meshless	meshless	ADJ
ejpam-4835	299	2	method	method	NOUN
ejpam-4835	299	3	based	base	VERB
ejpam-4835	299	4	on	on	ADP
ejpam-4835	299	5	rbfs	rbfs	NOUN
ejpam-4835	299	6	for	for	ADP
ejpam-4835	299	7	solving	solve	VERB
ejpam-4835	299	8	three	three	NUM
ejpam-4835	299	9	-	-	PUNCT
ejpam-4835	299	10	dimensional	dimensional	ADJ
ejpam-4835	299	11	multi	multi	ADJ
ejpam-4835	299	12	-	-	ADJ
ejpam-4835	299	13	term	term	ADJ
ejpam-4835	299	14	time	time	NOUN
ejpam-4835	299	15	fractional	fractional	ADJ
ejpam-4835	299	16	pdes	pde	NOUN
ejpam-4835	299	17	arising	arise	VERB
ejpam-4835	299	18	in	in	ADP
ejpam-4835	299	19	engineering	engineering	NOUN
ejpam-4835	299	20	phenomenons	phenomenon	NOUN
ejpam-4835	299	21	.	.	PUNCT
ejpam-4835	300	1	journal	journal	PROPN
ejpam-4835	300	2	of	of	ADP
ejpam-4835	300	3	king	king	PROPN
ejpam-4835	300	4	saud	saud	PROPN
ejpam-4835	300	5	universityscience	universityscience	PROPN
ejpam-4835	300	6	,	,	PUNCT
ejpam-4835	300	7	33(8):101604	33(8):101604	NUM
ejpam-4835	300	8	,	,	PUNCT
ejpam-4835	300	9	2021	2021	NUM
ejpam-4835	300	10	.	.	PUNCT
ejpam-4835	301	1	[	[	X
ejpam-4835	301	2	34	34	NUM
ejpam-4835	301	3	]	]	X
ejpam-4835	301	4	f	f	PROPN
ejpam-4835	301	5	wang	wang	PROPN
ejpam-4835	301	6	,	,	PUNCT
ejpam-4835	301	7	na	na	PROPN
ejpam-4835	301	8	ali	ali	PROPN
ejpam-4835	301	9	,	,	PUNCT
ejpam-4835	301	10	i	i	PROPN
ejpam-4835	301	11	ahmad	ahmad	PROPN
ejpam-4835	301	12	,	,	PUNCT
ejpam-4835	301	13	h	h	PROPN
ejpam-4835	301	14	ahmad	ahmad	PROPN
ejpam-4835	301	15	,	,	PUNCT
ejpam-4835	301	16	km	km	PROPN
ejpam-4835	301	17	alam	alam	PROPN
ejpam-4835	301	18	,	,	PUNCT
ejpam-4835	301	19	and	and	CCONJ
ejpam-4835	301	20	p	p	NOUN
ejpam-4835	301	21	thounthong	thounthong	NOUN
ejpam-4835	301	22	.	.	PUNCT
ejpam-4835	302	1	solution	solution	NOUN
ejpam-4835	302	2	of	of	ADP
ejpam-4835	302	3	burgers	burger	NOUN
ejpam-4835	302	4	’	'	PUNCT
ejpam-4835	302	5	equation	equation	NOUN
ejpam-4835	302	6	appears	appear	VERB
ejpam-4835	302	7	in	in	ADP
ejpam-4835	302	8	fluid	fluid	ADJ
ejpam-4835	302	9	mechanics	mechanic	NOUN
ejpam-4835	302	10	by	by	ADP
ejpam-4835	302	11	multistage	multistage	NOUN
ejpam-4835	302	12	optimal	optimal	ADJ
ejpam-4835	302	13	homotopy	homotopy	NOUN
ejpam-4835	302	14	asymptotic	asymptotic	ADJ
ejpam-4835	302	15	method	method	NOUN
ejpam-4835	302	16	.	.	PUNCT
ejpam-4835	303	1	thermal	thermal	ADJ
ejpam-4835	303	2	science	science	NOUN
ejpam-4835	303	3	,	,	PUNCT
ejpam-4835	303	4	26(1	26(1	NUM
ejpam-4835	303	5	part	part	NOUN
ejpam-4835	303	6	b):815–821	b):815–821	NOUN
ejpam-4835	303	7	,	,	PUNCT
ejpam-4835	303	8	2022	2022	NUM
ejpam-4835	303	9	.	.	PUNCT
ejpam-4835	304	1	[	[	X
ejpam-4835	304	2	35	35	NUM
ejpam-4835	304	3	]	]	X
ejpam-4835	304	4	f	f	PROPN
ejpam-4835	304	5	wang	wang	PROPN
ejpam-4835	304	6	,	,	PUNCT
ejpam-4835	304	7	e	e	PROPN
ejpam-4835	304	8	hou	hou	PROPN
ejpam-4835	304	9	,	,	PUNCT
ejpam-4835	304	10	i	i	PROPN
ejpam-4835	304	11	ahmad	ahmad	PROPN
ejpam-4835	304	12	,	,	PUNCT
ejpam-4835	304	13	h	h	PROPN
ejpam-4835	304	14	ahmad	ahmad	PROPN
ejpam-4835	304	15	,	,	PUNCT
ejpam-4835	304	16	and	and	CCONJ
ejpam-4835	304	17	y	y	PROPN
ejpam-4835	304	18	gu	gu	PROPN
ejpam-4835	304	19	.	.	PUNCT
ejpam-4835	305	1	an	an	DET
ejpam-4835	305	2	efficient	efficient	ADJ
ejpam-4835	305	3	meshless	meshless	ADJ
ejpam-4835	305	4	method	method	NOUN
ejpam-4835	305	5	for	for	ADP
ejpam-4835	305	6	hyperbolic	hyperbolic	ADJ
ejpam-4835	305	7	telegraph	telegraph	NOUN
ejpam-4835	305	8	equations	equation	NOUN
ejpam-4835	305	9	in	in	ADP
ejpam-4835	305	10	(	(	PUNCT
ejpam-4835	305	11	1	1	NUM
ejpam-4835	305	12	+	+	NUM
ejpam-4835	305	13	1	1	NUM
ejpam-4835	305	14	)	)	PUNCT
ejpam-4835	305	15	dimensions	dimension	NOUN
ejpam-4835	305	16	.	.	PUNCT
ejpam-4835	306	1	cmes	cmes	PROPN
ejpam-4835	306	2	-	-	PUNCT
ejpam-4835	306	3	computer	computer	NOUN
ejpam-4835	306	4	modeling	modeling	NOUN
ejpam-4835	306	5	in	in	ADP
ejpam-4835	306	6	engineering	engineering	NOUN
ejpam-4835	306	7	and	and	CCONJ
ejpam-4835	306	8	sciences	science	NOUN
ejpam-4835	306	9	,	,	PUNCT
ejpam-4835	306	10	128(2):687–98	128(2):687–98	NUM
ejpam-4835	306	11	,	,	PUNCT
ejpam-4835	306	12	2021	2021	NUM
ejpam-4835	306	13	.	.	PUNCT