SQU Journal for Science, 2019, 24(2), 109-121 DOI:10.24200/squjs.vol24iss2pp109-121 Sultan Qaboos University 109 Finite Element Convergence Analysis of a Schwarz Alternating Method for Nonlinear Elliptic PDEs Messaoud Boulbrachene Department of Mathematics, Sultan Qaboos University, P.O. Box 36, PC 123, Al- Khoud, Muscat Sultanate of Oman. Email: boulbrac@squ.edu.om. ABSTRACT: In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for nonlinear elliptic partial differential equations in the context of linear subdomain problems and nonmatching grids. The method stands on the combination of the convergence of linear Schwarz sequences with standard finite element L  -error estimate for linear problems. Keywords: Schwarz Method; Finite elements; Convergence. تحليل تقارب العناصر المحدودة لطريقة شوارز المتناوبة ألجهزة للمعادالت التفاضلية اإلخطية الجزئية بولبراشنمسعود لطريقة العناصر المحددة القياسية إلجراء تناوب شوارتز للمعادالت التفاضلية اإلخطية الجزئية في سياق في هذه الورقة ، نثبت التقارب الموحد :صلخمال وتقدير العناصر المحدودة لإلرهاب في تكمن الطريقة في الجمع بين التقارب بين تسلسل شوارز .مشاكل النطاق الفرعي الخطي والشبكات غير المتطابقة المشكالت الخطية. .، التقاربطريقة شوارز، العناصر المحدودة: مفتاحيةالكلمات ال 1. Introduction he Schwarz alternating method can be used to solve elliptic boundary value problems on domains that consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions that results from solving a sequence of elliptic boundary value problems in each of the subdomains. There has been extensive analysis of the Schwarz alternating method for nonlinear elliptic boundary value problems [1-4] and the references therein). Also, the effectiveness of Schwarz methods for these problems (especially those in fluid mechanics) has been demonstrated by many authors. In this paper, we are concerned with the finite element convergence analysis of overlapping Schwarz alternating methods in the context of nonmatching grids for nonlinear PDEs, where the Schwarz sub problems are linear. This study constitutes, to some extent, an improvement of the one achieved in [5], on a Schwarz method with nonlinear sub problems. For that, we develop an approach which combines the convergence result of Lui [6], with standard finite element error estimate for linear elliptic equations. For other works on finite element convergence analysis in the maximum norm of overlapping nonmatching Schwarz method, we refer to [7-12]. The rest of the paper is organized as follows. In section 2, we state the continuous alternating Schwarz sub problems and define their respective finite element counterparts in the context of nonmatching overlapping grids. In section 3, we give  L   - convergence analysis of the method. T BOULBRACHENE, M. 110 2. Preliminaries We begin by laying down some definitions and classical results related to linear elliptic equations. 2.1 Linear elliptic equations Let Ω be a bounded polyhedral domain of R²or R³ with sufficiently smooth boundary ∂Ω. We consider the bilinear form Ω ( , ) u vdxa u v    (1) the linear form  , = ( ) ( )f v f x v x dx   (2) the right hand side: f is a regular function, (3) the space     g 1= v such that = on ΩV H v g  (4) where g is a regular function defined on  . We consider the linear elliptic equation: Find ( )g V  such that      1 0 a( , )+c , , v v f v v H      (5) where such that , 0 0c R c c     (6) Let h V be the space of finite elements consisting of continuous piece-wise linear functions, , 1, 2,..., ( ) s s m h  be the basis functions of h V , and ( )m h denote the number of vertices of the triangulation in Ω. Let also 0 hV be the subspace of hV defined by   0 such that 0 on h h V v V v    (7) The discrete counterpart of (.,.) consists of finding g h h V  } such that     0 a( , )+c , , hh hv v f v v V     (8) where g h V is the space of     g such that= v = on Ω h h h V V v g  (9) and h  is the Lagrange interpolation operator on ∂Ω. Theorem 1. [13] Under suitable regularity of the solution of problem (5), there exists a constant c independent of h such that 2 ( ) ln h L Ch h      Lemma 1. [5] Let    1w H C    satisfy ( , ) ( , ) 0a w c w   ∀ non-negative φ∈  10H  , and 0w  on ∂Ω. Then 0w  on  . The proposition below establishes a Lipschitz continuous dependency of the solution with respect to the data. Notation 1. Let    , , , f g f g be a pair of data, and  ,f g  and  ,f g  be the corresponding solutions to (5). Proposition 1. Under the conditions of lemma 1, we have: ( )( ) ( ) 1 max , LL L f f g g                (10) Proof. First, set ( )( ) 1 max , LL f f g g            FINITE ELEMENT CONVERGENCE ANALYSIS OF A SCHWARZ ALTERNATING METHOD 111 Then ( )L f f f f      ( )L c f f f       ( )( ) 1 max , LL f c f f g g             f c  So      10a( , )+ , a( , ) , ( , ) 0, c c H                      a( + , )+c + , ,f c          On the other hand, we have 0 on       So ( , ) ( , ) 0 0 on a c                       Thus, making use of lemma 1, we get 0      On  Similarly, interchanging the roles of the couples    , , ,f g f g , we obtain 0 on       which completes the proof. Remark 1. Lemma 1 holds true in the discrete case. Indeed, assume that the discrete maximum principle (d.m.p) holds, i.e. the matrix resulting from the finite element discretization is an M-Matrix. Then we have: Lemma 2. Let h w V satisfy  , ,( 0 1, 2,..., () ) s s s ma c w hw       and 0w  on Ω . Then 0w  on  . Proof. The proof is a direct consequence of the discrete maximum principle. Let  ,f g and  ,f g be a pair of data and  ,h h f g  and  ,h h f g  be the corresponding solutions to problem (8). Proposition 2. Let the d.m.p hold. Then, under conditions of lemma 2, we have ( )( ) ( ) 1 max , h h LL L f f g g                Proof. The proof is similar to that of the continuous case. Indeed, as the basis functions s  > 0 of the space h V are positive, it suffices to make use of the discrete maximum principle. Let  ,f g and  ,f g be a pair of data and  ,h h f g  and  ,h h f g  be the corresponding solutions to problem (8). 3. Schwarz Alternating Methods for Nonlinear PDEs 3.1 The nonlinear PDE Consider the nonlinear PDE: Find  2u C  such that ( ) in 0 on u cu f u u        (11) BOULBRACHENE, M. 112 or in its weak form: Find  10 u H  such that        10, , ( ), a u v c u v f u v v H     (12) where (.)f is a non-decreasing nonlinearity We assume that (.)f is Lipschitz continuous, that is ( ) ( ) ,f x f y k x y x y    R such that k  where  is defined in (6). Theorem 2. [14] Under the above assumptions, Problem (11) has a unique solution. 3.2 The Linear Schwarz Sub problems We decompose Ω into two overlapping smooth subdomains 1  and 2  such that: 1 2     (13) We denote by i  the boundary of i  , i i j     , and i i    . We assume that the intersection of i and j ; i j is empty. Let 0 2 u be an initial guess. We define the alternating Schwarz sequences ( 1 1 n u  ) on 1  such that  1 21 1 n u C    solves 1 1 1 1 1 1 1 1 1 1 1 1 2 2 ( ) in 0 on on n n n n n n u cu f u u u u                (14) and the sequence  12 n u  such that  1 22 2 n u C    solves 1 1 2 2 2 2 1 2 2 1 1 2 1 2 ( ) in 0 on on n n n n n n u cu f u u u u                (15) Note that Schwarz subdomain problems (14) and (15) are linear. Theorem 3. [6] The sequences (14) and (15) converge uniformly in  2 1C  and   2 2 C  , respectively, to / , 1, 2 i i u u i   , where u is the solution of (11). 3.3 The variational Linear Schwarz Sub problems The corresponding variational problems read as follows: 1 1 1 n u V   solves      0 1 11 1 1 1 1 1 2 1 , , / / n n n n b u v f u v v V u u           (16) FINITE ELEMENT CONVERGENCE ANALYSIS OF A SCHWARZ ALTERNATING METHOD 113 and 1 2 2 n u V   solves      0 1 22 2 2 1 2 2 2 2 , , / / n n n n b u v f u v v V u u           (17) where  1i iV H  ;   0 1 0i i V H  (18)    , i i b u v u v cuv dx      (19)  , ( ) ( ) i i u v u x v x dx    (20) 3.4 The Discretization For 1, 2i  , let be i h  be a standard regular and quasi-uniform finite element triangulation in i  ; i h , being the mesh size. The two meshes being mutually independent on 1 2   , a triangle belonging to one triangulation does not necessarily belong to the other. Let us define the discrete analog of spaces i V and 0 iV , respectively, that is  1( )such that / ii i h h K V v C v P K        0 such that 0 on i i i h h V Vv v    and let ih  denote the Lagrange interpolation operator on i  The discrete Maximum principle (see [15,16]). We assume that the respective matrices resulting from the discretization of problems (16) and (17) are M-matrices. 3.5 The Finite Element Linear Schwarz Sub problems Let 2 0 h u be the discrete analog of 0 2 u that is,   2 2 0 0 2h h u r u where 2h r denotes the finite element interpolation operator in Ω. We define the sequence   1 1n h u  such that 1 1 1n hh u V   solves        1 1 1 1 1 2 0 1 1 1 1 , , / n n hh h n n h h h b u v f u v v V u u            (21) and the sequence ( 2 1n h u  ) such that 2 2 1n hh u V   solves        2 2 2 2 2 1 0 1 2 1 1 2 2 , , / / n n hh h n n h h h b u v f u v v V u u              (22) 4. L  - Convergence Analysis This section is devoted to the proof of the main result of the present paper. To that end, we begin by introducing two discrete auxiliary Schwarz sequences and prove a fundamental lemma. BOULBRACHENE, M. 114 4.1 Auxiliary Discrete Schwarz Sub problems We construct a sequence   1 1n h   such that 11 1n hh V   solves        1 1 1 2 1 1 1 0 1 1 1 1 , , / / n h n n h h h n b v f u v v V u u               (23) and the sequence   2 1n h   such that 22 1n hh V   solves        2 2 2 1 2 2 1 0 2 1 1 2 2 , , / / h n h h n n n h b v f u v v V u u               (24) Then, it is clear that 1 1n h   and 2 1n h   are the finite element approximation of 1 1n u  and 1 2 n u  defined in (16) and (17), respectively. Notation 2. From now on, we shall adopt the following notations:    1 21 2. , .L L      (25)      1 21 2. , . , .L L L          (26) 1 2h h h     (27) 4.2 The Main Result The following lemma will play a key role in proving the main result of this paper. Lemma 3. 1 1 1 1 1 2 2 1 1 2 0 0 n n n n i i i i h h h i i u u u u           2 2 1 1 2 2 2 1 2 0 0 n n n n i i i i h h h i i u u u u          Proof. The proof will be carried out by induction. For n=1, we have in 1  . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 h h h h u u u u      and, making use of Proposition 2, we obtain 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 2 1 1 1 1 1 max ( ) ( ) , h h h h h h u u u f u f u u u               1 1 0 0 0 0 1 1 1 1 2 2 1 1 2 max , h h h k u u u u u            We then have to distinguish between two cases 0 0 0 0 0 0 1 1 2 2 1 1 1 2 1 1 : max , h h h k k u u u u u u            FINITE ELEMENT CONVERGENCE ANALYSIS OF A SCHWARZ ALTERNATING METHOD 115 and 0 0 0 0 0 0 1 1 2 2 2 2 1 2 2 2 : max , h h h k u u u u u u           Case 1. implies that 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 h h h k u u u u u       and hence 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 h h h u u u u u     Case 2. implies that 1 1 1 1 0 0 1 1 1 1 2 2 1 1 2 h h h u u u u u     and, in both cases, we have 1 1 1 1 0 0 0 0 1 1 1 1 1 1 2 2 1 1 1 2 h h h h u u u u u u u       (28) Similarly, we have in 2  1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 h h h h u u u u      1 1 0 0 1 1 2 2 2 2 1 1 2 2 2 max , h h h h h k u u u u u              1 1 0 0 1 1 2 2 2 2 1 1 2 2 1 max , h h h k u u u u u            Here also we need to consider the following two cases: 1: 0 0 1 1 0 0 2 2 1 1 2 2 2 1 2 max , h h h k k u u u u u u            2: 0 0 1 1 1 1 2 2 1 1 1 1 2 1 1 max , h h h k u u u u u u           Case 3. implies that 1 1 1 1 0 0 2 2 2 2 2 2 2 2 2 h h h k u u u u u       so 1 1 1 1 0 0 2 2 2 2 2 2 2 2 2 h h h u u u u u     BOULBRACHENE, M. 116 Case 4. implies that 1 1 1 1 1 1 2 2 2 2 1 1 2 2 1 h h h u u u u u     1 1 1 1 0 0 0 0 2 2 1 1 1 1 2 2 2 1 1 2 h h h h u u u u u u         Thus, in both cases, we have 1 1 1 1 1 1 0 0 0 0 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 h h h h h u u u u u u u u          (29) For n = 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 h h h h u u u u      2 2 1 1 1 1 1 1 1 1 2 2 1 1 1 1 max ( ) ( ) , h h h h h u f u f u u u              2 2 1 1 1 1 1 1 1 1 2 2 1 1 2 max , h h h k u u u u u            Case 1. 1 1 1 1 1 1 1 1 2 2 1 1 1 2 1 1 max , h h h k u u u u u u            2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 h h h k u u u u u       2 2 1 1 1 1 1 1 1 1 h h u u u    2 2 1 1 0 0 0 0 1 1 1 1 1 1 2 2 1 1 1 2 h h h h u u u u u u         Case 2. 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2 1 max , h h h u u u u u u           2 2 2 2 1 1 1 1 0 0 0 0 1 1 1 1 2 2 1 1 1 1 2 2 1 1 2 1 1 2 h h h h h h u u u u u u u u u             So in both cases 2 2 2 2 1 1 1 1 0 0 0 0 1 1 1 1 2 2 1 1 1 1 2 2 1 1 2 1 1 2 h h h h h h u u u u u u u u u             (30) or FINITE ELEMENT CONVERGENCE ANALYSIS OF A SCHWARZ ALTERNATING METHOD 117 2 1 2 2 1 1 1 1 2 2 1 1 2 0 0 i i i i h h h i i u u u u          (31) On the other hand 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 h h h h u u u u      2 2 1 1 2 2 2 2 2 2 1 1 2 2 2 max , h h h k u u u u u            Case 1. 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 h h h k u u u u u       2 2 1 1 2 2 2 2 2 2 h h u u u    2 2 1 1 1 1 0 0 0 0 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 h h h h h u u u u u u u            Case 2. 2 2 2 2 2 2 2 2 2 2 1 1 2 2 1 h h h u u u u u     2 2 2 2 1 1 1 1 0 0 0 0 2 2 1 1 2 2 1 1 1 1 2 2 2 1 2 1 1 2 h h h h h h u u u u u u u u               So in both cases 2 2 2 2 2 2 1 1 1 1 0 0 0 0 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 2 1 1 2 h h h h h h h u u u u u u u u u u                or 2 2 2 2 2 2 1 1 2 2 2 1 2 0 0 i i i i h h h i i u u u u          (32) Now assume that 1 1 1 1 1 2 2 2 1 2 0 0 n n n n i i i i h h h i i u u u u           and 2 2 1 1 2 2 2 1 2 0 0 n n n n i i i i h h h i i u u u u          and let us prove that 1 1 1 1 1 1 1 2 2 1 1 2 0 0 n n n n i i i i h h h i i u u u u             and 1 1 1 1 2 2 1 1 2 2 2 1 2 0 0 n n n n i i i i h h h i i u u u u              BOULBRACHENE, M. 118 Indeed, we have 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n n n n n n h h h h u u u u             1 1 1 1 1 1 2 2 1 1 1 max , n n n n n n h h h h h k u u u u u                1 1 1 1 1 1 2 2 1 1 2 max , n n n n n n h h h k u u u u u              and, as above, we need to distinguish between two cases: 1: 1 1 2 2 1 1 1 2 1 max , n n n n n n h h h k k u u u u u u            and 2: 1 1 2 2 2 2 1 2 2 max , n n n n n n h h h k u u u u u u           Case 1. implies that 1 1 1 1 1 1 1 1 1 1 1 1 1 n n n n n n h h h k u u u u u           1 1 1 1 1 1 1 1 n n n n h h u u u       1 1 1 1 1 1 1 2 2 1 1 2 0 0 n n n n i i i i h h h i i u u u               while Case 2. implies that 1 1 1 1 1 1 1 1 2 2 1 1 2 n n n n n n h h h u u u u u          1 1 1 1 1 1 2 2 1 1 2 0 0 n n n n i i i i h h h i i u u u              So in both cases 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 0 0 n n n n n n i i i i h h h h i i u u u u u                 1 1 1 2 2 1 2 0 0 n n i i i i h h i i u u          Likewise 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 n n n n n n h h h u u u u u            1 1 1 1 2 2 2 2 1 2 2 2 2 max , n n n n h h h h h k u u u u u                1 1 1 1 2 2 2 2 1 1 2 2 1 max , n n n n n n h h h k u u u u u               FINITE ELEMENT CONVERGENCE ANALYSIS OF A SCHWARZ ALTERNATING METHOD 119 Here also we need to discuss two cases: Case 1: 1 1 2 2 1 1 2 2 2 1 2 max , n n n n n n h h h k k u u u u u u             implies 1 1 1 1 2 2 2 2 2 2 2 2 2 n n n n n n h h h u u u u u          1 1 2 2 1 1 2 2 2 1 2 0 0 n n n n i i i i h h h i i u u u              Case 2: 1 1 1 1 2 2 1 1 1 1 2 1 1 max , n n n n n n h h h k u u u u u u              implies 1 1 1 1 1 1 2 2 2 2 1 1 2 2 1 n n n n n n h h h u u u u u            1 1 1 1 2 2 1 1 1 1 2 2 2 1 1 2 0 0 n n n n n n i i i i h h h h i i u u u u                   Then, in both cases, we have 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 0 0 n n n n n n n n i i i i h h h h h i i u u u u u u                      1 1 1 1 2 2 1 2 0 0 n n i i i i h h i i u u           which completes the proof. Theorem 4. There exists n h > 0 with lim 0 n n h   , such that  1 1 lim 0, 1, 2 i i i h Ln u u i      Proof. Let us give the proof for i=1, the case i=2 being similar. Indeed, as 1 1 1 1 1 2 2 1 1 2 0 0 n n n n i i i i h h h i i u u u u           and 2 1 1 1 ln n n h u Ch h  2 2 2 2 ln n n h u Ch h  Then,   2 1 1 1 2 1 ln n n h u u n Ch h   (33) BOULBRACHENE, M. 120 Also 1 1 1 1 1 1 1 1 1 n n n n n h h u u u u u u     Let ε>0. Theorem 3 implies that there exists 0 n N such that, ∀ 0 n n 1 1 1 2 n u u    Taking account of (33), the Theorem follows by choosing n h > 0 such that   2 0 ln , 2 1 n n h h n n n C      Conclusion We have proved convergence of the standard finite element approximation for alternating Schwarz procedure in the context of nonmatching grids. Other type of discretizations may also be considered like mixing finite elements and finite differences. Also, the knowledge of a rate of convergence of the Schwarz procedure will enable derivation of error estimate, in each subdomain, between the discrete Schwarz sequence and the exact solution of the nonlinear PDE. Conflict of interest The author declares no conflict of interest. Acknowlegement The author would like to thank Sultan Qaboos University for providing financial support to attend and present the paper at International Conference on Mathematics (ICOMATH2018), Istanbul, 3-6 July, 2018, Turkey. References 1. Proceedings of the 1st International Symposium on Domain Decomposition for Partial Differential Equations, Editors, Glowinski, R., Meurant, G., Golub, G.H., S.I.A.M, Phildelphia, USA, 1998. 2. Lions, P.L. On the Schwarz Alternating Method I, Proceedings of the1st, International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM Philadelphia, USA, 1988, 1-41. 3. Lions, P.L. On the Schwarz Alternating Method II, Proceedings of the 2nd International Symposium on on Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia, USA, 1989, 47-70. 4. Lui, S-H. On Monotone and Schwarz Alternating Methods For Nonlinear Elliptic PDEs. Mathematical Modelling and Numerical Analysis, (ESAIM:M2AN), 2001, 35, 1, 1-15. 5. Harbi, A. and Boulbrachene, M. Maximum Norm Analysis of a Nonmatching Grids Method for Nonlinear Elliptic PDEs. Journal of Applied Mathematics, Article ID 605140, 2011, 1-18. 6. Lui, S-H. On Linear Monotone Iteration and Schwarz Methods For Nonlinear Elliptic PDEs. Numerishe Mathematik, 2002, 109-129. 7. Boulbrachene, M., Cortey-Dumont, P. and Miellou, J.C. Mixing Finite Elements and Finite Differences on a Subdomain Decomposition Method, Proceedings of the 2nd International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, USA, 1988, 198-216. 8. Boulbrachene, M. and Saadi, S. Maximum Norm Analysis of an Overlapping Nonmatching Grids Method For the Obstacle Problem, Advances in Difference Equations, 2006, 1-10. 9. Harbi, A. Maximum Norm Analysis of An Arbitrary Number of Nonmatching Grids Method for Nonlinear Elliptic PDEs, Journal of Applied Mathematics. Article ID 895182, 2013, 1-21. 10. Boulbrachene, M. and Al-Farei, Q. Maximum Norm Error Analysis of a Grids Method Nonmatching grids finite element method for Linear PDEs. Applied Mathematics and Computation, 2014, 238, 21-29. 11. Harbi, A. Maximum Norm Analysis of a Nonmatching Method for a Class of Variational Inequalities with Nonlinear Source Terms. Journal of Inequalities and Applications, 2016, 181, 1-27. 12. Bahi, M., Boulaares, S. and Haiour, M. The Maximum Norm Analysis of a Nonmatching Grids Method for a Class of Parabolic Equations with Nonlinear Source Terms, Applied Sciences, 2018, 20, 1-17. FINITE ELEMENT CONVERGENCE ANALYSIS OF A SCHWARZ ALTERNATING METHOD 121 13. Nitsche, J. L  - Convergence of Finite Element Approximations, Mathematical aspects of finite element methods, Lecture Notes in Mathematics, 1977, 606, 261-274. 14. Pao, C.V. Nonlinear Parabolic and Elliptic Equations, Plenium Press, New York, 1992.. 15. Ciarlet, P.G. and Raviart, P.A. Maximum principle and uniform convergence for the finite element method, Computer Methods in Applied Mechanics and Engineering, 1973, 2, 17-31. 16. Karatson, J. and Korotov, S. Discrete maximum principle for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numerische Mathematik, 2005, 99, 669-698. Received 6 December 2018 Accepted 13 May 2019