Int. J. Anal. Appl. (2023), 21:71

L∞-Convergence Analysis of a Finite Element Linear Schwarz Alternating Method for

a Class of Semi-Linear Elliptic PDEs

Qais Al Farei, Messaoud Boulbrachene∗

Department of Mathematics, Sultan Qaboos University, P.O. Box 36, Muscat 123, Oman

∗Corresponding author: boulbrac@squ.edu.om

Abstract. In this paper, we prove uniform convergence of the standard finite element method for a

Schwarz alternating procedure for a class of semi-linear elliptic partial differential equations, in the

context of linear iterations and non-matching grids. More precisely, making use of the subsolution-

based concept, we prove that finite element Schwarz iterations converge, in the maximum norm, to the

true solution of the PDE. We also give numerical results to validate the theory. This work introduces

a new approach and generalizes the one in [14] as it encompasses a larger class of problems.

1. INTRODUCTION

The Schwarz alternating method can be used to solve elliptic boundary value problems on domains

which consist of two or more overlapping subdomains. The solution is approximated by an infinite

sequence of functions which results from solving a sequence of elliptic boundary value problems in each

of the subdomains. The literature in this area is huge and one can refer to [2], [3] and to proceedings

of the annual International Symposium on Domain Decomposition for Partial Differential Equations,

starting from [1].

The mathematical analysis of Schwarz alternating method for nonlinear elliptic boundary value

problems has been extensively studied in the last three decades (c.f., e.g., [2], [3], [5], [6] and the

references therein).

On the numerical analysis side and, more specifically, non-matching grid discretizations, to the best

of our knowledge, only few works are known in the literature regarding the convergence and error

estimates analysis for discrete Schwarz procedures (c.f. [7], [8], [9], [10], [12], [15]).

Received: Jan. 3, 2023.

2020 Mathematics Subject Classification. 65N30, 65N15.

Key words and phrases. Schwarz method; finite elements; non-matching grids; subsolutions; uniform convergence.

https://doi.org/10.28924/2291-8639-21-2023-71
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-71


2 Int. J. Anal. Appl. (2023), 21:71

The main motivation in using non-matching grid discretizations resides in their flexibility as they

can be applied to solve many practical problems which cannot be handled by global discretizations

because of the complexity of the domain’s geometry. They allow the choice of different mesh sizes

and different orders of approximate polynomials in different subdomains according to the different

properties of the solution and different requirements of the practical problems.

In the present paper, we are interested in a non-matching grid finite element approximation method

for the class of PDEs 
 −∆u = f (x,u) in Ωu = g on ∂Ω (1.1)

where Ω ⊂Rd,d = 2, 3 is a bounded domain with boundary ∂Ω, ∆ is the Laplace operator, f (.) is a
smooth nonlinearity, and g is a regular function defined on ∂Ω.

To be more specific, let Ω = Ω1 ∪ Ω2 such that Ω1 ∩ Ω2 6= ∅, γi = ∂Ωi∩ Ωj, Γi = ∂Ωi ∩∂Ω and
∂Ωi ; i = 1, 2, the boundary of Ωi. Let also c(x) be a positive smooth function. Then following the

work of S.H.Lui [6], given initial smooth guesses u01 and u
0
2, we approximate the solution of problem

(1.1) by Schwarz sequences
(
uni
)
such that un1 ∈ C

2
(

Ω̄1
)
, n ≥ 1 solves the linear subproblem



−∆un1 + cu

n
1 = f (u

n−1
1 ) + cu

n−1
1 in Ω1

un1 = u
n−1
2 on γ1

un1 = g on Γ1

(1.2)

on Ω1,and un2 ∈ C
2
(

Ω̄2
)
, n ≥ 1 solves the linear subproblem

−∆un2 + cu

n
2 = f (u

n−1
2 ) + cu

n−1
2 in Ω2

un2 = u
n
1 on γ2

un2 = g on Γ2

(1.3)

on Ω2.

In this paper, motivated by the uniform convergence result [6],

lim
n→∞

‖uni −u‖L∞(Ωi ) = 0, i = 1, 2,

we prove that the corresponding finite elements Schwarz sequences (un1h1 ) and (u
n
2h2

), generated in

the context of non-matching grids, converge, in the maximum norm, to the exact solution of problem

(1.1). That is,

lim
n→∞

∥∥u −unihi∥∥L∞(Ωi ) = 0, i = 1, 2,
where, hi is the mesh-size on Ωi, and ui = u/Ωi .



Int. J. Anal. Appl. (2023), 21:71 3

To that end, by means of the concept of subsolutions, we establish a fundamental lemma which

consists of estimating the error, at each iteration, between the continuous and the discrete Schwarz

iterations, on each subdomain.

This work introduces a new approach and uses weaker assumptions on the nonlinearity than the

one developed in [14] to derive the convergence result.

The layout of the paper is as follows. In section 2, we recall some standard results related to linear

elliptic boundary problems. In section 3, we recall the existence of a solution for the nonlinear PDE,

and define both the continuous and discrete variational formulations of subproblems (1.2) and (1.3).

In section 4, we prove the main results of this paper. Finally, in section 5, we give some numerical

results to validate the theory.

2. PRELIMINARIES

We begin by recalling some definitions and classical results related to linear elliptic equations.

2.1. Linear Elliptic equations. We introduce the bilinear form

a(ξ,v) =

∫
Ω

(∇ξ.∇v + cξv)dx ∀v ∈ H1 (Ω) , (2.1)

the linear form

(f ,v) =

∫
Ω

f (x).v(x)dx ∀v ∈ H1 (Ω) , (2.2)

where the right hand side

f is a regular function, (2.3)

and the space

V(g) = {v ∈ H1 (Ω) such that v = g on ∂Ω}, (2.4)

where g is a regular function defined on ∂Ω. Note that V̊ = H10 (Ω).

We consider the linear elliptic equation: Find ξ ∈V(g) such that

a (ξ,v) = (f ,v) , ∀v ∈ V̊(Ω) (2.5)

Lemma 2.1. [6] (Weak maximum principle) Let w ∈ H1 (Ω) ∩ C(Ω̄) satisfy a(w,φ) ≥ 0 ∀ non-
negative φ ∈ V̊, and w ≥ 0 on ∂Ω. Then w ≥ 0 on Ω̄.

Definition 2.1. A function ξ̌ ∈ H1 (Ω) is a subsolution of (2.5) if{
a(ξ̌,v) ≤ (f ,v) ∀v ≥ 0,v ∈ V̊(Ω)
ξ̌ ≤ g

(2.6)

Definition 2.2. A function ξ̂ ∈ H1 (Ω) is a supersolution of (2.5) if{
a(ξ̂,v) ≥ (f ,v) ∀v ≥ 0,v ∈ V̊(Ω)
ξ̂ ≥ g

(2.7)



4 Int. J. Anal. Appl. (2023), 21:71

Lemma 2.2. The solution ξ of (2.5) is the least upper bound of the set of subsolutions.

Proof. (2.6) can be re-written as

a(−ξ̌,v) ≥ (−f ,v) ∀v ≥ 0,v ∈ V̊.

Subtracting this result from (2.5) yields

a(ξ− ξ̌,v) ≥ 0 ∀v ≥ 0,v ∈ V̊.

Since ξ− ξ̌ ≥ 0 on ∂Ω, it follows from lemma 2.1 that ξ̌ ≤ ξ on Ω̄, which completes the proof. �

The proposition below establishes a continuous Lipschitz property of the solution with respect to

the data.

Notation 2.1. Let (f ,g) and (f̃ , g̃) be a pair of data, and ξ = ∂ (f ,g) and ξ̃ = ∂( f̃ , g̃) be the

corresponding solutions to (2.5).

Proposition 2.1. [9] Let β be a positive constant such that c/β ≥ 1. Let also lemma 2.1 hold. Then,

∥∥ξ− ξ̃∥∥
L∞(Ω)

≤ max
{

1

β

∥∥f − f̃∥∥
L∞(Ω)

;‖g − g̃‖L∞(∂Ω)
}

(2.8)

Let Vh be the space of finite elements consisting of continuous piece-wise linear functions, φs,
s = 1, 2, ...,m(h) be the basis functions of Vh. Let also V̊h be the subspace of Vh defined by

V̊h = {v ∈Vh such that v = 0 on ∂Ω} (2.9)

The discrete counterpart of (2.5) consists of finding ξh ∈V
(g)
h

such that

a(ξh,v) = (f ,v) ∀v ∈ V̊h (2.10)

where

V(g)
h

= {v ∈Vh such that v = πhg on ∂Ω }, (2.11)

and πh is the linear Lagrange interpolation operator on ∂Ω.

The discrete version of lemma 2.1 stays true provided a discrete maximum principle (d.m.p) holds

(the matrix resulting from the finite element discretization is an M-matrix). See [16].

Lemma 2.3. Let wh ∈Vh satisfy a(wh,φs) ≥ 0 ∀s = 1, 2, ...,m(h) and wh ≥ 0 on ∂Ω. Then, under
the d.m.p, we have wh ≥ 0 on Ω̄.

Definition 2.3. A function ξ̌h ∈Vh is a subsolution of (2.10) if
 a(ξ̌h,φs) ≤ (f ,φs) ∀φs ≥ 0,∀s = 1, 2, ...,m(h)
ξ̌h ≤ πhg

(2.12)



Int. J. Anal. Appl. (2023), 21:71 5

Definition 2.4. A function ξ̂h ∈Vh is a supersolution of (2.10) if
 a(ξ̂h,φs) ≥ (f ,φs) ∀φs ≥ 0,∀s = 1, 2, ...,m(h)ξ̂h ≥ πhg (2.13)

Lemma 2.4. Let the d.m.p hold. Then the solution ξh of (2.10) is the least upper bound of the set

of subsolutions.

Proof. The proof is similar to that of lemma 2.2. Indeed, as φs ≥ 0 are non-negative, it suffices to
make use of lemma 2.3. �

Now we give the finite element counterpart of proposition 2.1.

Notation 2.2. Let (f ,g) and (f̃ , g̃) be a pair of data, with ξh = ∂h(f ,g) and ξ̃h = ∂h(f̃ , g̃) be the

corresponding discrete solutions to (2.10).

Proposition 2.2. [9] Let β be a positive constant such that c/β ≥ 1. Then, under the d.m.p and
conditions of lemma 2.3, we have∥∥ξh − ξ̃h∥∥L∞(Ω) ≤ max

{
1

β

∥∥f − f̃∥∥
L∞(Ω)

;‖g − g̃‖L∞(∂Ω)
}

(2.14)

Finally, we recall a standard maximum norm error estimate [18].

Theorem 2.1. [18] Under suitable regularity of the solution of problem (2.5), there exists a constant

C independent of h such that

‖ξ−ξh‖L∞(Ω) ≤ Ch
2 |ln h|

3. SCHWARZ METHOD FOR NONLINEAR PDEs

We first recall the following classical existence result due to Pao [4].

3.1. The Nonlinear PDE. We shall consider the following nonlinear PDE: Find u ∈ C2(Ω) such that


 −∆u = f (x,u) in Ω
u = g on ∂Ω

(3.1)

For the sake of convenience, we will suppress the dependence of the space variable x.

Definition 3.1. [4] A function ǔ ∈ C2(Ω) is a subsolution of (3.1) if
 −∆ǔ ≤ f (ǔ) in Ωǔ ≤ g on ∂Ω (3.2)



6 Int. J. Anal. Appl. (2023), 21:71

Definition 3.2. [4] A function û ∈ C2(Ω) is a supersolution of (3.1) if


 −∆û ≥ f (û) in Ω
û ≥ g on ∂Ω

(3.3)

Suppose that (3.1) has a subsolution ǔ and a supersolution û such that ǔ ≤ û on Ω. Define the
sector

A = {u ∈ C2(Ω̄); ǔ ≤ u ≤ û on Ω̄}. (3.4)

Assume that

−c (u −v) ≤ f (u) − f (v) ∀v ≤ u ∈A (3.5)

Then, thanks to Pao [4], (3.1) has a solution (not necessarily unique) in A.

Theorem 3.1. [6] Let u02 = ǔ on Ω̄; i = 1, 2, with ǔ = 0 on ∂Ω. Define the linear Schwarz sequences

generated by the subproblems (1.2) and (1.3).Then uni → u in C
2(Ω̄i ), where u is a solution of (3.1)

in A. Similarly, if u02 = û on Ω̄ with û = 0 on ∂Ω instead, then the same conclusion holds.

3.2. Continuous variational Schwarz subproblems. The weak form of (1.2) and (1.3) read as

follows: find un1 ∈ H
1 (Ω) such that:



a1(u

n
1,v) = (F (u

n−1
1 ),v) ∀v ∈ V̊1

un1 = u
n−1
2 on γ1

un1 = g on Γ1,

(3.6)

and un2 ∈ H
1 (Ω) such that



a2(u

n
2,v) = (F (u

n−1
2 ),v) ∀v ∈ V̊2

un2 = u
n
1 on γ2

un2 = g on Γ2

(3.7)

respectively, where

ai (ui,v) =

∫
Ωi

(∇ui∇v + cuiv)dx, (3.8)

and

(F (ui ),v) =

∫
Ωi

(f (ui ) + cui )vdx ; i = 1, 2. (3.9)



Int. J. Anal. Appl. (2023), 21:71 7

3.3. Finite element discretization. Let τhi ; i = 1, 2 be a standard quasi-uniform regular finite ele-
ment triangulation on Ωi ; hi being its mesh size. We introduce the finite element spaces Vhi and V̊hi
as follows:

Vhi = {v ∈ C
0(Ω̄i ) : v/K ∈ P1 ∀K ∈τhi}, (3.10)

and

V̊hi = {v ∈Vhi : v = 0 on Γi}, (3.11)

where P1 denotes the space of linear polynomials on K ∈τhi, with degree ≤ 1. The two meshes are
also assumed to be overlaping and non-matching in the sense that they are mutually independent on

the overlap region.

The discrete maximum principle (d.m.p). We assume that the meshing on each subdomain

satisfies the discrete maximum principle. In other words, the matrices resulting from the discretization

of (3.6) and (3.7) are M-matrices.

3.4. Discrete variational Schwarz subproblems. Let u0ihi be the discrete analog of u
0
i , i.e.; u

0
ihi

=

rhi (u
0
i ), where rhi denotes the finite element Lagrange interpolation operator on Ωi. Now, we define

the discrete Schwarz sequences (un1h1 ) such that u
n
1h1
∈Vh1 solves



a1(u

n
1h1
,v) = (F (un−1

1h1
),v) ∀v ∈ V̊h1

un1h1 = πh1 (u
n−1
2h2

) on γ1

un1h1 = πhg on Γ1,

(3.12)

and
(
un2h2

)
such that un2h2 ∈Vh2 solves



a2(u

n
2h2
,v) = (F (un−1

2h2
),v) ∀v ∈ V̊h2

un2h2 = πh2 (u
n
1h1

) on γ2

un2h2 = πhg on Γ2,

(3.13)

where πhi denotes the Lagrange interpolation operator on γi. Below, we construct a finite element

discretization of subproblems (3.12) and (3.13), as in Figure 1, using a quasi-uniform regular finite

element triangulation on both subdomains as stated before.



8 Int. J. Anal. Appl. (2023), 21:71

Figure 1. A sample of two overlapping nonmatching grids.

4. L∞- CONVERGENCE ANALYSIS

This section is devoted to the proof of the main results of the present paper. We first introduce

two continuous and two discrete auxiliary Schwarz sequences and prove a fundamental lemma.

4.1. Continuous auxiliary Schwarz subproblems. For ũ0i = u
0
i ; i = 1, 2, we define the continuous

auxiliary Schwarz sequence (ũn1 ) such that ũ
n
1 ∈V1 solves


a1(ũ

n
1,v) = (F (u

n−1
1h1

),v) ∀v ∈ V̊1

ũn1 = πh1 (u
n−1
2h2

) on γ1

ũn1 = πhg on Γ1

(4.1)

and
(
ũn2
)
such that ũn2 ∈V2 solves



a2(ũ

n
2,v) = (F (u

n−1
2h2

),v) ∀v ∈ V̊2

ũn2 = πh2 (u
n
1h1

) on γ2

ũn2 = πhg on Γ2

(4.2)

where un1h1 and u
n
2h2

are the Schwarz iterates defined in (3.12) and (3.13), respectively.



Int. J. Anal. Appl. (2023), 21:71 9

4.2. Discrete Auxiliary Schwarz subproblems. Likewise, for ũ0ihi = u
0
ihi

; i = 1, 2, we define the

discrete auxiliary Schwarz sequences (ũn1h1 ) such that ũ
n
1h1
∈Vh1 solves


a1(ũ

n
1h1
,v) = (F (un−11 ),v) ∀v ∈ V̊h1

ũn1h1 = πh1 (u
n−1
2 ) on γ1

ũn1h1 = πhg on Γ1

(4.3)

and
(
ũn2h2

)
such that ũn2h2 ∈Vh2 solves


a2(ũ

n
2h2
,v) = (F (un−12 ),v) ∀v ∈ V̊h2

ũn2h2 = πh2 (u
n
1 ) on γ2

ũn2h2 = πhg on Γ2

(4.4)

where un1 and u
n
2 are the Schwarz iterates defined in (3.6) and (3.7), respectively.

Notation 4.1. From now onward, we shall adopt the following notations:

C is ageneric constant independent of h and n,

‖.‖1 = ‖.‖L∞(Ω1) ; |.|1 = ‖.‖L∞(γ1) ,

‖.‖2 = ‖.‖L∞(Ω2) ; |.|2 = ‖.‖L∞(γ2) ,

πh1 = πh2 = πh,

and

h = max
i=1,2

hi.

Lemma 4.1. Assume that

max
{
‖ũni ‖W 2,p(Ωi ) , ‖u

n
i ‖W 2,p(Ωi )

}
≤ C.

Then, we have ∥∥ũni −unihi∥∥L∞(Ωi ) ≤ Ch2 |ln h| , (4.5)
∥∥uni − ũnihi∥∥L∞(Ωi ) ≤ Ch2 |ln h| . (4.6)

where C is a constant independent of both hi ; i = 1, 2 and n.

Proof. It is clear that unihi and ũ
n
ihi

are the discrete counterparts of ũni and u
n
i , respectively. So, as

the latter are both uniformly bounded in W 2,p(Ωi ), the desired error estimates follows from Theorem

2.1. �



10 Int. J. Anal. Appl. (2023), 21:71

4.3. The main results. The following lemma plays a crucial role in deriving the main result of this

paper.

Lemma 4.2. Assume that f (.) is a Lipschitz continuous function, i.e., there is a constant k > 0 such

that

|f (x) − f (y)| ≤ k |x −y| ∀x,y ∈R. (4.7)

Then,

‖un1 −u
n
1h‖1 ≤ (2n)Ch

2 |ln h| (4.8)

and

‖un2 −u
n
2h‖2 ≤ (2n + 1)Ch

2 |ln h| . (4.9)

Remark 4.1. Note that the assumption k/β ≤ 1 used in [14] is no longer needed in this paper.

Proof. The proof will be carried out by induction. Also, for the sake of simplicity, we shall ignore the

boundary condition on Γi ; i = 1, 2.

Indeed, on Ω1, problem (4.1) for n = 1 reads as follows
 a1(ũ

1
1,v) = (F (u

0
1h),v) ∀v ∈ V̊1

ũ11 = πh(u
0
2h) on γ1.

(4.10)

As ũ11 is also a subsolution for (4.10), we have
 a1(ũ

1
1,v) ≤ (F (u

0
1h),v) ∀v ∈ V̊1,v ≥ 0

ũ11 ≤ πh(u
0
2h) on γ1.

But 
 a1(ũ

1
1,v) ≤ (F (u

0
1h) −F (u

0
1 ) + F (u

0
1 ),v) ∀v ∈ V̊1,v ≥ 0

ũ11 ≤ πh(u
0
2h) −πh(u

0
2 ) + πh(u

0
2 ) on γ1,

then, since F (.) is Lipschitz continuous and γ1 ⊂ Ω2, this implies
 a1(ũ

1
1,v) ≤ (C

∥∥u01 −u01h∥∥1 + F (u01 ),v) ∀v ∈ V̊1,v ≥ 0
ũ11 ≤

∥∥u02 −u02h∥∥2 + πh(u02 ) on γ1.
Then, making use of standard uniform estimate, we have∥∥u0i − rh(u0i )∥∥i ≤ Ch2 |ln h| ; i = 1, 2. (4.11)
Hence 


a1(ũ

1
1,v) ≤ (F (u

0
1 ) + Ch

2 |ln h| ,v) ∀v ∈ V̊1,v ≥ 0

ũ11 ≤ πh(u
0
2 ) + Ch

2 |ln h| on γ1



Int. J. Anal. Appl. (2023), 21:71 11

Let Ũ11 be the solution of the problem with source term F (u
0
1 ) +Ch

2 |ln h| and boundary data πh(u02 ) +
Ch2 |ln h| . That is,

Ũ11 = ∂(F (u
0
1 ) + Ch

2 |ln h| , πh(u02 ) + Ch
2 |ln h|)

Then, as

u11 = ∂(F (u
0
1 ) , u

0
2 ),

making use of proposition 2.1, yields∥∥Ũ11 −u11∥∥1 ≤ max {Ch2 |ln h| ; Ch2 |ln h|}
≤ Ch2 |ln h| .

Hence, due to lemma 2.2, we have

ũ11 ≤ Ũ
1
1 ≤ u

1
1 + Ch

2 |ln h| .

Putting

α11 = ũ
1
1 −Ch

2 |ln h| ,

we get

α11 ≤ u
1
1. (4.12)

and using (4.5), for n = 1, we also get∥∥ũ11 −u11h∥∥1 ≤ Ch2 |ln h| .
Thus, ∥∥α11 −u11h∥∥1 = ∥∥ũ11 −Ch2 |ln h|−u11h∥∥1 (4.13)

≤ Ch2 |ln h| + Ch2 |ln h|

≤ 2Ch2 |ln h| .

Now, consider problem (4.3) for n = 1:
 a1(ũ

1
1h,v) = (F (u

0
1 ),v) ∀v ∈ V̊1h

ũ11h = πh(u
0
2 ) on γ1.

(4.14)

As ũ11h is also a subsolution for (4.14), we have

a1(ũ

1
1h,φs) ≤ (F (u

0
1 ),φs) ∀φs ≥ 0,∀s

ũ11h ≤ πh(u
0
2 ) on γ1,

which implies 

a1(ũ

1
1h,φs) ≤ (F (u

0
1 ) −F (u

0
1h) + F (u

0
1h),φs) ∀φs ≥ 0,∀s

ũ11h ≤ πh(u
0
2 ) −πh(u

0
2h) + πh(u

0
2h) on γ1.



12 Int. J. Anal. Appl. (2023), 21:71

Since F (.) and πh are Lipschitz, we get
 a1(ũ

1
1h,φs) ≤ (C

∥∥u01 −u01h∥∥1 + F (u01h),φs) ∀φs ≥ 0,∀s
ũ11h ≤

∥∥ u02 −u02h∥∥2 + πh(u02h) on γ1.
Hence, using (4.11), yields

 a1(ũ
1
1h,φs) ≤ (F (u

0
1h) + Ch

2 |ln h| ,φs) ∀φs ≥ 0,∀s

ũ11h ≤ πh(u
0
2h) + Ch

2 |ln h| on γ1.

Let Ũ11h be the solution of the problem with source term F (u
0
1h) + Ch

2 |ln h| and boundary data
πh(u

0
2h) + Ch

2 |ln h|, that is,

Ũ11h = ∂h(F (u
0
1h) + Ch

2 |ln h| , πh(u02h) + Ch
2 |ln h|)

Then, as

u11h = ∂h(F (u
0
1h) , πh(u

0
2h)),

making use of proposition 2.2, yields

∥∥Ũ11h −u11h∥∥1 ≤ max {Ch2 |ln h| ; Ch2 |ln h|}
≤ Ch2 |ln h| ,

and due to lemma 2.4, we have

ũ11h ≤ Ũ
1
1h ≤ u

1
1h + Ch

2 |ln h| .

Now, putting

α11h = ũ
1
1h −Ch

2 |ln h| ,

it follows that

α11h ≤ u
1
1h. (4.15)

And making use of (4.6) for n = 1, we get

∥∥u11 − ũ11h∥∥1 ≤ Ch2 |ln h| .
Thus,

∥∥α11h −u11∥∥1 = ∥∥ũ11h −Ch2 |ln h|−u11∥∥1 . (4.16)
≤ Ch2 |ln h| + Ch2 |ln h|

≤ 2Ch2 |ln h|



Int. J. Anal. Appl. (2023), 21:71 13

Now, combining (4.12), (4.13), (4.15) and (4.16), we get

u11 ≤ α
1
1h + 2Ch

2 |ln h|

≤ u11h + 2Ch
2 |ln h|

≤ α11 + 4Ch
2 |ln h|

≤ u11 + 4Ch
2 |ln h| .

That is, ∥∥u11 −u11h∥∥1 ≤ 2Ch2 |ln h| . (4.17)
Similarly on Ω2, for n = 1 in (4.2), we have

 a2(ũ
1
2,v) = (F (u

0
2h),v) ∀v ∈ V̊2

ũ12 = πh(u
1
1h) on γ2.

(4.18)

The solution ũ12 is also a subsolution for (4.18). That is,

a2(ũ

1
2,v) ≤ (F (u

0
2h),v) ∀v ∈ V̊2,v ≥ 0

ũ12 ≤ πh(u
1
1h) on γ2,

or 

a2(ũ

1
2,v) ≤ (F (u

0
2h,v) −F (u

0
2,v) + F (u

0
2 ),v) ∀v ∈ V̊2,v ≥ 0

ũ12 ≤ πh(u
1
1h) −πh(u

1
1 ) + πh(u

1
1 ) on γ2.

As F (.) is Lipschitz continuous function and γ2 ⊂ Ω1, this implies
 a2(ũ

1
2,v) ≤ (

∥∥u02 −u02h∥∥2 + F (u02 ),v) ∀v ∈ V̊2,v ≥ 0
ũ12 ≤

∥∥u11 −u11h∥∥1 + πh(u11 ) on γ2
Using (4.11) and the resulting estimate (4.17), we obtain

 a2(ũ
1
2,v) ≤ (F (u

0
2 ) + Ch

2 |ln h| ,v) ∀v ∈ V̊2,v ≥ 0

ũ12 ≤ πh(u
1
1 ) + 2Ch

2 |ln h| on γ2.

Let Ũ12 be the solution of the equation with source term F (u
0
2 ) +Ch

2 |ln h| and boundary data πh(u11 ) +
2Ch2 |ln h| , that is,

Ũ12 = ∂(F (u
0
2 ) + Ch

2 |ln h| , πh(u11 ) + 2Ch
2 |ln h|).

Then, as

u12 = ∂(F (u
0
2 ) , u

1
1 ),

making use of proposition 2.1, we get∥∥Ũ12 −u12∥∥2 ≤ max {Ch2 |ln h| ; 2Ch2 |ln h|}
≤ 2Ch2 |ln h| .



14 Int. J. Anal. Appl. (2023), 21:71

Also, due to lemma 2.2, we have

ũ12 ≤ Ũ
1
2 ≤ u

1
2 + 2Ch

2 |ln h|

Now, putting

α12 = ũ
1
2 − 2Ch

2 |ln h| ,

yields

α12 ≤ u
1
2. (4.19)

And due to (4.5) for n = 1, we have ∥∥ũ12 −u12h∥∥2 ≤ Ch2 |ln h| .
Thus, it follows that ∥∥α12 −u12h∥∥2 = ∥∥ũ12 − 2Ch2 |ln h|−u12h∥∥2 (4.20)

≤
∥∥ũ12 −u12h∥∥1 + 2Ch2 |ln h|

≤ 3Ch2 |ln h| .

Again, on Ω2, for n = 1 in (4.4), we have
 a2(ũ

1
2h,v) = (F (u

0
2 ),v) ∀v ∈ V̊2h

ũ12h = πh(u
1
1 ) on γ2.

(4.21)

The solution ũ12h being also a subsolution, we have
 a1(ũ

1
2h,φs) ≤ (F (u

0
2 ),φs) ∀φs ≥ 0,∀s

ũ12h ≤ πh(u
1
1 ) on γ2.

Then, as F (.) and πh are Lipschitz, we get
 a1(ũ

1
2h,φs) ≤ (C

∥∥u02 −u02h∥∥2 + F (u02h),φs) ∀φs ≥ 0,∀s
ũ12h ≤

∥∥u11 −u11h∥∥1 + πh(u11h) on γ2,
or 

 a1(ũ
1
2h,φs) ≤ (F (u

0
2h) + Ch

2 |ln h| ,φs) ∀φs ≥ 0

ũ12h ≤ πh(u
1
1h) + 2Ch

2 |ln h| on γ2.

Hence, ũ12h is a subsolution for the problem with source term F (u
0
2h) + Ch

2 |ln h| and boundary term
πh(u

1
1h) + 2Ch

2 |ln h|. Let Ũ12h be the solution of such a problem, that is,

Ũ12h = ∂h(F (u
0
2h) + Ch

2 |ln h| , πh(u11h) + 2Ch
2 |ln h|)

Then, we have

ũ12h ≤ Ũ
1
2h.



Int. J. Anal. Appl. (2023), 21:71 15

As

u12h = ∂h(F (u
0
2h) , πh(u

1
1h)),

making use of proposition 2.2, we get∥∥Ũ12h −u12h∥∥2 ≤ max {Ch2 |ln h| ; 2Ch2 |ln h|}
≤ 2Ch2 |ln h| .

So, due to lemma 2.4, we have

ũ12h ≤ Ũ
1
2h ≤ u

1
2h + 2Ch

2 |ln h|

Now, putting

α12h = ũ
1
2h − 2Ch

2 |ln h| ,

yields

α12h ≤ u
1
2h. (4.22)

And making use of (4.6) for n = 1, we get∥∥α12h −u12∥∥2 = ∥∥ũ12h − 2Ch2 |ln h|−u12∥∥2 (4.23)
(4.24)

≤ 3Ch2 |ln h| .

Now, combining statements (4.19), (4.20), (4.22) and (4.23), we obtain

u12 ≤ α
1
2h + 3Ch

2 |ln h|

≤ u12h + 3Ch
2 |ln h|

≤ α12 + 6Ch
2 |ln h|

≤ u12 + 6Ch
2 |ln h| .

That is, ∥∥u12 −u12h∥∥2 ≤ 3Ch2 |ln h| . (4.25)
Now, for n = 2 on Ω1, (4.1) reads

 a1(ũ
2
1,v) = (F (u

1
1h),v) ∀v ∈ V̊1

ũ21 = πh(u
1
2h) on γ1.

(4.26)

As ũ21 is also a subsolution, we have
 a1(ũ

2
1,v) ≤ (F (u

1
1h),v) ∀v ∈ V̊1,v ≥ 0

ũ21 ≤ πh(u
1
2h) on γ1.



16 Int. J. Anal. Appl. (2023), 21:71

And, since F (.) is Lipschitz, we get

a1(ũ

2
1,v) ≤ (F (u

1
1 ) + 2Ch

2 |ln h| ,v) ∀v ∈ V̊1

ũ21 ≤ πh(u
1
2 ) + 3Ch

2 |ln h| on γ1.

So, ũ21 is a subsolution for the problem with source term F (u
1
1 ) + 2Ch

2 |ln h| and boundary term
πh(u

1
2 ) + 3Ch

2 |ln h|. Let Ũ21 be the solution of such a problem. That is,

Ũ21 = ∂(F (u
1
1 ) + 2Ch

2 |ln h| , πh(u12 ) + 3Ch
2 |ln h|).

Then, due to lemma 2.2, we have

ũ21 ≤ Ũ
2
1.

Furthermore, as

u21 = ∂(F (u
1
1 ) , u

1
2 ),

making use of proposition 2.1, we get∥∥Ũ21 −u21∥∥1 ≤ max {2Ch2 |ln h| ; 3Ch2 |ln h|}
≤ 3Ch2 |ln h| .

Hence,

ũ21 ≤ Ũ
2
1 ≤ u

2
1 + 3Ch

2 |ln h| .

Putting

α21 = ũ
2
1 − 3Ch

2 |ln h| ,

yields

α21 ≤ u
2
1. (4.27)

Making use of (4.5) for n = 2, we get∥∥α21 −u21h∥∥1 = ∥∥ũ21 − 3Ch2 |ln h|−u21h∥∥1 (4.28)
≤ Ch2 |ln h| + 3Ch2 |ln h|

≤ 4Ch2 |ln h| .

Now for n = 2 on Ω1, (4.3) reads
 a1(ũ

2
1h,v) = (F (u

1
1 ),v) ∀v ∈ V̊1h

ũ21h = πh(u
1
2 ) on γ1.

(4.29)

As ũ21h is also a subsolution, we have
 a1(ũ

2
1h,φs) ≤ (F (u

1
1 ),φs) ∀φs ≥ 0,∀s

ũ21h ≤ πh(u
1
2 ) on γ1.



Int. J. Anal. Appl. (2023), 21:71 17

Similarly, as above, this implies
 a1(ũ

2
1h,φs) ≤ (F (u

1
1h) + 2Ch

2 |ln h| ,φs) ∀φs ≥ 0,∀s

ũ21h ≤ πh(u
1
2h) + 3Ch

2 |ln h| on γ1.

And, due to lemma 2.4,

ũ21h ≤ Ũ
2
1h = ∂h(F (u

1
1h) + 2Ch

2 |ln h| , πh(u12h) + 3Ch
2 |ln h|)

But

u21h = ∂h(F (u
1
1h) , πh(u

1
2h))

Then, using proposition 2.2, yields the estimate∥∥Ũ21h −u21h∥∥1 ≤ max {2Ch2 |ln h| ; 3Ch2 |ln h|}
≤ 3Ch2 |ln h| .

Hence

ũ21h ≤ Ũ
2
1h ≤ u

2
1h + 3Ch

2 |ln h| .

Putting

α21h = ũ
2
1h − 3Ch

2 |ln h| ,

it follows that

α21h ≤ u
2
1h, (4.30)

and, (4.6) for n = 2, implies that∥∥α21h −u21∥∥1 = ∥∥ũ21h − 3Ch2 |ln h|−u21∥∥1 (4.31)
≤ 4Ch2 |ln h| .

Combining (4.27), (4.28), (4.30) and (4.31), we obtain

u21 ≤ α
2
1h + 4Ch

2 |ln h|

≤ u21h + 4Ch
2 |ln h|

≤ α21 + 8Ch
2 |ln h|

≤ u21 + 8Ch
2 |ln h| .

Thus, ∥∥u21 −u21h∥∥1 ≤ 4Ch2 |ln h| . (4.32)
Similarly, for n = 2 on Ω2,(4.2) we have


a2(ũ

2
2,v) = (F (u

1
2h),v) ∀v ∈ V̊2

ũ22 = πh(u
2
1h) on γ2.



18 Int. J. Anal. Appl. (2023), 21:71

Using a similar argument as above, one can prove that ũ22 is a subsolution for the problem with source

term F (u12 ) + 3Ch
2 |ln h| and boundary condition πh(u21 ) + 4Ch

2 |ln h|. Let Ũ22 be the solution of such
a problem, that is

Ũ22 = ∂(F (u
1
2 ) + 3Ch

2 |ln h| , πh(u21 ) + 4Ch
2 |ln h|),

Then, as

u22 = ∂(F (u
1
2 ) , u

2
1 ),

making use of proposition 2.1, we get∥∥Ũ22 −u22∥∥2 ≤ 4Ch2 |ln h| .
Putting

α22 = ũ
2
2 − 4Ch

2 |ln h| ,

we obtain

α22 ≤ u
2
2, (4.33)

and, making use of (4.5) for n = 2, we get∥∥α22 −u22h∥∥2 = ∥∥ũ22 − 4Ch2 |ln h|−u22h∥∥1 (4.34)
≤ 5Ch2 |ln h| .

Likewise, for n = 2 on Ω2, we can also establish that

α22h ≤ u
2
2h (4.35)

and ∥∥α22h −u22∥∥2 ≤ 5Ch2 |ln h| (4.36)
Hence, combining (4.33), (4.34), (4.35) and (4.36), we obtain∥∥u22 −u22h∥∥2 ≤ 5Ch2 |ln h| . (4.37)

Now, let us assume that (4.8) and (4.9) hold. We need to prove it for the (n + 1)th step. Indeed,

consider the problem 
 a1(ũ

n+1
1 ,v) = (F (u

n
1h),v) ∀v ∈ V̊1

ũn+11 = πh(u
n
2h) on γ1.

Then, we also have 
 a1(ũ

n+1
1 ,v) ≤ (F (u

n
1h),v) ∀v ∈ V̊1,v ≥ 0

ũn+11 ≤ πh(u
n
2h) on γ1,

which can be rewritten as

a1(ũ

n+1
1 ,v) ≤ (F (u

n
1h) −F (u

n
1 ) + F (u

n
1 ),v) ∀v ∈ V̊1,v ≥ 0

ũn+11 ≤ πh(u
n
2h) −πh(u

n
2 ) + πh(u

n
2 ) on γ1.



Int. J. Anal. Appl. (2023), 21:71 19

Since F (.) is Lipschitz continuous, γ1 ⊂ Ω2, this implies that

a1(ũ

n+1
1 ,v) ≤ (F (u

n
1 ) + (2n)Ch

2 |ln h| ,v) ∀v ∈ V̊1,v ≥ 0

ũn+11 ≤ πh(u
n
2 ) + (2n + 1)Ch

2 |ln h| on γ1.

This means that ũn+11 is a subsolution for the problem with source term F (u
n
1 ) + (2n)Ch

2 |ln h| and
boundary term πh(un2 ) + (2n + 1)Ch

2 |ln h| . Let Ũn+11 be the solution of such a problem. That is,

Ũn+11 = ∂
(
F (un1 ) + (2n)Ch

2 |ln h| , πh(un2 ) + (2n + 1)Ch
2 |ln h|

)
.

Then, making use of lemma 2.2, we have

ũn+11 ≤ Ũ
n+1
1 = ∂(F (u

n
1 ) + (2n)Ch

2 |ln h| , πh(un2 ) + (2n + 1)Ch
2 |ln h|)

But

un+11 = ∂(F (u
n
1 ) , u

n
2 ),

then, making use of proposition 2.1, we have∥∥Ũn+11 −un+11 ∥∥1 ≤ (2n + 1)Ch2 |ln h| ,
and due to lemma 2.2,

ũn+11 ≤ Ũ
n+1
1 ≤ u

n+1
1 + (2n + 1)Ch

2 |ln h| .

Putting

αn+11 = ũ
n+1
1 − (2n + 1)Ch

2 |ln h| , (4.38)

yields

αn+11 ≤ u
n+1
1 .

And, using (4.5), we get ∥∥αn+11 −un+11h ∥∥1 ≤ (2n + 1)Ch2 |ln h| . (4.39)
The solution of (4.3) is also a subsolution:

 a1(ũ
n+1
1h

,φs) ≤ (F (un1 ),φs) ∀φs ≥ 0,∀s

ũn+1
1h
≤ πh(un2 ) on γ1,

which, in turn, can be rewritten as

a1(ũ

n+1
1h

,φs) ≤ (F (un1 ) −F (u
n
1h) + F (u

n
1h),φs) ∀φs ≥ 0,∀s

ũn+1
1h
≤ πh(un2 ) −πh(u

n
2h) + πh(u

n
2h) on γ1.

Since F (.) is Lipschitz continuous, γ1 ⊂ Ω2, this implies

a1(ũ

n+1
1h

,φs) ≤ (F (un1h) + (2n)Ch
2 |ln h| ,φs) ∀φs ≥ 0,∀s

ũn+1
1h
≤ πh(un2h) + (2n + 1)Ch

2 |ln h| on γ1.



20 Int. J. Anal. Appl. (2023), 21:71

In other words, ũn+1
1h

is a subsolution for the problem with data F (un1h) + (2n)Ch
2 |ln h| and πh(un2h) +

(2n + 1)Ch2 |ln h| . Let Ũn+1
1h

be the solution of such a problem. That is,

Ũn+11h = ∂h(F (u
n
1h) + (2n)Ch

2 |ln h| , πh(un2h) + (2n + 1)Ch
2 |ln h|).

But, as

un+11h = ∂h(F (u
n
1h) , πh(u

n
2h)),

making of proposition 2.2, we get∥∥Ũn+11h −un+11h ∥∥1 ≤ (2n + 1)Ch2 |ln h| .
And so, due to lemma 2.4, we have

ũn+11h ≤ Ũ
n+1
1h ≤ u

n+1
1h + (2n + 1)Ch

2 |ln h| .

Now putting

αn+11h = ũ
n+1
1h − (2n + 1)Ch

2 |ln h| , (4.40)

and using (4.6), we obtain ∥∥αn+11h −un+11 ∥∥1 ≤ 2(n + 1)Ch2 |ln h| . (4.41)
Hence, similarly to above, combining (4.38), (4.39), (4.40) and (4.41), we obtain∥∥un+11 −un+11h ∥∥1 ≤ 2(n + 1)Ch2 |ln h| . (4.42)
Which is the desired result in Ω1.

Likewise, the estimate for the iterate n + 1 in Ω2 can be proved using similar arguments as above,

which yields ∥∥un+12 −un+12h ∥∥1 ≤ (2n + 3)Ch2 |ln h| .
�

Corollary 4.1. ∀n ≥ 1 fixed, we have

lim
h→0
‖uni −u

n
ih‖i = 0, i = 1, 2.

Proof. The proof is straightforward. For fixed n ≥ 1, passing to the limit as h → 0 to both (4.8) and
(4.9), the corollary follows on both subdomains.

�

Now, we are in position to prove the following convergence result:

Corollary 4.2. There exists hn > 0 with hn → 0, such that

lim
n→∞

∥∥ui −unihn∥∥i = 0; i = 1, 2. (4.43)



Int. J. Anal. Appl. (2023), 21:71 21

Proof. Let us give the proof of (4.43) on Ω1, the one on Ω2 is similar. We know that

‖u1 −un1h‖1 ≤‖u1 −u
n
1‖1 + ‖u

n
1 −u

n
1h‖1

Letting � > 0, Theorem 3.1 implies that there exists N ∈N such that

‖u1 −un1‖1 ≤
�

2
∀n > N

Hence, due to (4.8), we have

‖un1 −u
n
1h‖1 ≤ (2n)Ch

2 |ln h| ,

Thus, the convergence result follows by choosing hn > 0 such that

h2n |ln hn| ≤
�

4Cn
∀n > N.

�

5. Numerical Experiments

In this section, we conduct numerical experiments on two model problems to validate the theory.

The first model is chosen so that it does not have an exact solution, while we know an exact solution

for the second one. For both models, we adopt the following notations:

• hi ; i = 1, 2, are the mesh sizes of the triangulations in Ωi.
• δ is the size of overlap between both subdomains.

• ERRORhi =
∥∥∥u −unihi∥∥∥i is the maximum error between the exact solution u and the discrete

Schwarz iterate on each subdomain.

We shall conduct the two tests to investigate the behavior of ERRORhi as follows:

(1) We fix the mesh sizes hi and vary the number of Schwarz iterations n,

(2) We fix the number of iterations n and vary the mesh sizes hi,

(3) We consider the sequence of mesh sizes hi,n as n varies.

For all the experiments, we consider Ω = [0, 1] × [0, 1] . The "FreeFEM++" software, see [19], is
adapted to obtain the numerical results for both models.

5.1. First example: In this example, we consider the boundary value problem

−∆u =

−σu
1 + au + bu2

in Ω

u = x + y on ∂Ω

(5.1)

where σ = 1,a = 0 and b = 0.25. This problem describes the enzyme kinetics model with inhibition.

The value of the constant c is defined by determining suitable lower and upper solutions to the problem



22 Int. J. Anal. Appl. (2023), 21:71

(5.1) satisfying Definitions (3.2) and (3.3), respectively. The sector 〈ǔ, û〉 = 〈0, 12〉 is taken and c is
evaluated by [4]:

c = max

{
−
∂f

∂u
; x ∈ Ω̄, ǔ ≤ u ≤ û

}
. (5.2)

For c = 1, f (u) satisfies the one-sided Lipshitz condition

f (u1) − f (u2) ≥−(u1 −u2) for ǔ ≤ u2 ≤ u1 ≤ û. (5.3)

A unique positive solution for the above model is also ensured in the sector 〈0, 2〉, (see [4]). Since
it is difficult to obtain the exact solution for the problem, we use a P2− finite element approximation
of the exact solution of the problem on Ω, instead. Figure 2 represents the solution u of the above

problem using a uniform fixed mesh size of 1
30
.

Figure 2. P2−Approximate solution

We divide Ω into two overlapping non-matching subdomains Ω1 and Ω2 such that each subdomain

is independently discretized into a quasi-uniform mesh with P1 triangular elements and different mesh

size hi ; i = 1, 2. In order for the maximum principle to be satisfied here, we construct a triangulation

with acute angles for every K ∈τhi ; i = 1, 2, using a Delaunay triangulation algorithm.

When both mesh sizes are fixed to be h1 =
1

32
and h2 =

1
24

with two sizes of the overlap δ = 1
8

and δ = 1
4
, the approximated solution of the 35th iterate are represented in Figure 3, respectively.

This shows the convergence of Schwarz sequences unihi to u. The same result can be easily shown for
different mesh refinements.

Furthermore, Tables 1 and 2 represent the approximate solution values at some points in the domain
with a stopping criterion ε = e−5 for both subdomains. The obtained information of the tables
show the monotone convergence of the Schwarz sequences unihi , where u1h1 and u2h2 are the P2-
approximation of the nonlinear PDE problem on Ω1 and Ω2, respectively. It is also seen that the
number of Schwarz iterations decreases as the overlap size increases.



Int. J. Anal. Appl. (2023), 21:71 23

Table 1. Approximate solution values at some points when δ = 1
8

n u1h1 (
1
4
, 1

2
) u1h1 (

1
2
, 1

2
) u2h2 (

2
3
, 1

4
) u2h2 (

3
4
, 2

3
)

0 0 0 0 0

1 0.345889 0.0817873 0.642487 1.1308

2 0.572373 0.536428 0.814077 1.32854

3 0.689899 0.809101 0.890129 1.40256

4 0.745107 0.943227 0.926039 1.43526

5 0.770443 1.00566 0.942779 1.44996

Table 2. Approximate solution values at some points when δ = 1
4

n u1h1 (
1
4
, 1

2
) u1h1 (

1
2
, 1

2
) u2h2 (

2
3
, 1

4
) u2h2 (

3
4
, 2

3
)

0 0 0 0 0

1 0.390914 0.16399 0.741398 1.23655

2 0.677421 0.800115 0.902063 1.41263

3 0.762939 0.992884 0.943188 1.45016

4 0.784635 1.04217 0.953611 1.45921

(a) (b)

(c) (d)

Figure 3. Iterative process of the first example on both Ω1 and Ω2. Approximate

solution at 35th iteration when (A)δ = 1
8
and (B) δ = 1

4
. Maximum errors versus

number of iterations (C) and versus meshsizes (D).



24 Int. J. Anal. Appl. (2023), 21:71

In the first place, when we put h1 =
1

32
, h2 =

1
24

with both δ = 1
8
and δ = 1

4
as before, and vary the

number of Schwarz iterations over 1 ≤ n ≤ 35, one can observe how the maximum errors decrease as
the number of Schwarz iterations and the overlap size increase.

Next, we fix the number of iterations to be n = 8 when δ = 1
8
, n = 5 when δ = 1

4
and vary the mesh

sizes to be h1 =
1

4×2N , h2 =
1

3×2N when 1 ≤ N ≤ 5, instead. One can notice that the maximum
errors decrease as the mesh sizes get smaller. Also, the bigger overlap size is, the smaller errors and

the closer the curves are. Figure 3 shows both plots of ERRORhi .

5.2. Second example: We consider the following problem

−∆u = σup in Ω

u = 12
(x+y+1)

2 on ∂Ω

(5.4)

where σ = −1 and p = 2. This problem describes the concentration of free atoms in the dissociation
process. One can verify that the function f (u) satisfies the one-sided Lipschitz condition (5.3). Also,

the value of c satisfying (5.2) is 24. Hence, there exists a positive solution for the model in the sector

〈0, 12〉 . Furthermore, one can verify that the exact solution of the model is given by

u =
12

(x + y + 1)
2

The exact solution u is represented in Figure 4 using a uniform fixed mesh size of 1
30
.

Figure 4. Exact solution

In this example, we build the same triangulation as in the first experiment in order to satisfy the

maximum principle. We also examine the performance of the iterative approach for different values of

the number of iterations, with only overlap size of δ = 1
8
, by doing a similar analysis to the one made

in the first example. The numerical results are shown in Figure 5, where the first and second figures

represent the approximate solution at the initial and 35th iteration, while the third and fourth ones

display the relationship of maximum errors with number of iterations and mesh sizes, respectively.



Int. J. Anal. Appl. (2023), 21:71 25

Moreover, the approximated solution values at some certain points in the domain with the same

stopping criterion as in the first example for both subdomains are represented in Table 3. We notice

the monotone convergence of the Schwarz sequences unihi .

(a) (b)

(c) (d)

Figure 5. Iterative process of the second example on both Ω1 and Ω2. (A)

Approximate solution at first iteration. (B) Approximate solution at 35th iteration.

(C) Maximum errors versus number of iterations with fixed mesh sizes. (D)Maximum

errors versus meshsizes with fixed number of iterations.

Table 3. Approximate solution values at some points when δ = 1
8

n u1h1 (
1
4
, 1

2
) u1h1 (

1
2
, 1

2
) u2h2 (

2
3
, 1

4
) u2h2 (

3
4
, 2

3
)

0 0 0 0 0

1 3.31901 0.709984 2.71408 1.64465

2 3.685 2.14238 3.13503 1.93498

3 3.86836 2.7503 3.23898 2.02499

4 3.90591 2.9386 3.26161 2.04795

5 3.9165 2.98647 3.2672 2.05378

6 3.91889 2.99829 3.26853 2.0552

7 3.9195 3.00117 3.26885 2.05555



26 Int. J. Anal. Appl. (2023), 21:71

We conclude this section by validating the convergence result (Corollary 4.2). Applying the context

of it to both examples with mesh sizes of h1,n =
1

2n+1
and h2,n =

1
3n+1

on both subdomains when

1 ≤ n ≤ 35, we see in Figure 6 that

∥∥∥ui −unihi,n∥∥∥i ≤ 6n2 ; i = 1, 2,∀n
In particular, the asymptotic behavior of our iterative approach is indicated to be at least O( 1

n2
). This

proves that our numerical results are in agreement with our theory.

(a) (b)

(c) (d)

Figure 6. Plots of the maximum errors for both examples by considering the

meshsize sequences hi,n for 1 ≤ n ≤ 35. (A) First maximum error for Example 1. (B)
Second maximum error for Example 1. (C) First maximum error for Example 2. (D)

Second maximum error for Example 2.



Int. J. Anal. Appl. (2023), 21:71 27

6. CONCLUSION

In this paper, we have proved the convergence of the standard finite element approximation of

monotone linear Schwarz alternating procedure for a class of semilinear elliptic PDEs, in the context

of non-matching grids. In order to prove the main result, we used the concept of subsolutions to

estimate, at each Schwarz iteration, the gap between the continuous and approximated Schwarz

sequences. We have also conducted numerical experiments to show the agreement with the theory.

We believe that the availability of a rate of convergence of the Schwarz procedure will help to derive

an error estimate between the discrete Schwarz sequence and the exact solution of the semilinear PDE

on each subdomain.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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https://doi.org/10.1007/s10598-022-09558-x
https://doi.org/10.1016/0045-7825(73)90019-4

	1. INTRODUCTION
	2. PRELIMINARIES
	2.1. Linear Elliptic equations

	3. SCHWARZ METHOD FOR NONLINEAR PDEs
	3.1. The Nonlinear PDE
	3.2. Continuous variational Schwarz subproblems
	3.3. Finite element discretization
	3.4. Discrete variational Schwarz subproblems

	4. L- CONVERGENCE ANALYSIS
	4.1. Continuous auxiliary Schwarz subproblems
	4.2. Discrete Auxiliary Schwarz subproblems
	4.3. The main results

	5. Numerical Experiments
	5.1. First example:
	5.2. Second example:

	6. CONCLUSION
	References