Maximum norm error analysis of a nonmatching grids finite element method for linear elliptic PDEs q Messaoud Boulbrachene a,⇑ , Qais Al Farei b a Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Muscat 123, Oman b Higher College of Technology, IT Department, Muscat 123, Oman article info Keywords: Elliptic PDEs Schwarz alternating method Nonmatching grids Finite element L 1 – error estimate abstract In this paper, we study a nonmatching grid finite element approximation of linear elliptic PDEs in the context of the Schwarz alternating domain decomposition.We show that the approximation converges optimally in the maximum norm, on each subdomain, making use of the geometrical convergence of both the continuous and corresponding discrete Schwarz sequences. We also give some numerical results to support the theory. Ó 2014 Elsevier Inc. All rights reserved. 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. Extensive analysis of Schwarz alternating method for linear elliptic boundary value problems can be found in [1,2].The effectiveness of Schwarz methods for these problems, especially those in fluid mechanics, has been demonstrated in many papers. For that, we refer to the proceedings of the annual domain decomposition conference beginning with [3]. In this paper, we are interested in the error analysis in the maximum norm of a finite element Schwarz alternating method for linear elliptic problems on two overlapping subdomains with nonmatching grids: we consider a domain X which is the union of two overlapping subdomains where each subdomain has its own triangulation. This kind of discretization is very inter- esting as they can be applied to solving many practical problems which cannot be handled by global discretizations.They are earning particular attention of computational experts and engineers as 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. For other kinds of discretization of elliptic PDEs, we refer to [4–6]. More specifically, we show that the approximation converges optimally on each subdomain, developing a new approach which consists of combining the geometrical convergence of both the continuous and discrete Schwarz sequences with an estimate in the maximum norm between the continuous and discrete Schwarz iterates. More precisely, if ðu n i; Þ denotes the Schwarz sequence in the subdomain X i and ðu n i;h i Þ is its finite element counterpart with respect to the triangulation with meshsize h i , we show that u n i  u n i;h i       L 1 ðX i Þ 6 Ch 2 log h j j ð1:1Þ http://dx.doi.org/10.1016/j.amc.2014.03.146 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. q The authors would like to thank the referee for his careful reading and valuable comments. ⇑ Corresponding author. E-mail addresses: boulbrac@squ.edu.om (M. Boulbrachene), qais.alfarei@hct.edu.om (Q. Al Farei). Applied Mathematics and Computation 238 (2014) 21–29 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc