Abstract
We provide a maximum norm analysis of an overlapping Schwarz method on nonmatching grids for second-order elliptic obstacle problem. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The grid points on the subdomain boundaries need not match the grid points from the other subdomain. Under a discrete maximum principle, we show that the discretization on each subdomain converges quasi-optimally in the L∞ norm.
Similar content being viewed by others
References
Badea L: On the Schwarz alternating method with more than two subdomains for nonlinear monotone problems. SIAM Journal on Numerical Analysis 1991,28(1):179–204. 10.1137/0728010
Boulbrachene M, Cortey-Dumont Ph, Miellou J-C: Mixing finite elements and finite differences in a subdomain method. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987). SIAM, Philadelphia; 1988:198–216.
Cai X-C, Mathew TP, Sarkis MV: Maximum norm analysis of overlapping nonmatching grid discretizations of elliptic equations. SIAM Journal on Numerical Analysis 2000,37(5):1709–1728. 10.1137/S0036142998348741
Ciarlet PG, Raviart P-A: Maximum principle and uniform convergence for the finite element method. Computer Methods in Applied Mechanics and Engineering 1973,2(1):17–31. 10.1016/0045-7825(73)90019-4
Cortey-Dumont Ph: On finite element approximation in the L∞-norm of variational inequalities. Numerische Mathematik 1985,47(1):45–57. 10.1007/BF01389875
Kuznetsov YuA, Neittaanmäki P, Tarvainen P: Overlapping domain decomposition methods for the obstacle problem. In Domain Decomposition Methods in Science and Engineering (Como, 1992), Contemp. Math.. Volume 157. Edited by: Quarteroni A, Périaux J, Kuznetsov YuA, Widlund OB. American Mathematical Society, Rhode Island; 1994:271–277.
Kuznetsov YuA, Neittaanmäki P, Tarvainen P: Schwarz methods for obstacle problems with convection-diffusion operators. In Domain Decomposition Methods in Scientific and Engineering Computing (University Park, Pa, 1993), Contemp. Math.. Volume 180. Edited by: Keyes DE, Xu J. American Mathematical Society, Rhode Island; 1994:251–256.
Lions P-L: On the Schwarz alternating method. I. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987). SIAM, Philadelphia; 1988:1–42.
Lions P-L: On the Schwarz alternating method. II. Stochastic interpretation and order properties. In Domain Decomposition Methods (Los Angeles, Calif, 1988). SIAM, Philadelphia; 1989:47–70.
Mathew TP, Russo G: Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids. Mathematics of Computation 2003,72(242):619–656.
Zeng J, Zhou S: On monotone and geometric convergence of Schwarz methods for two-sided obstacle problems. SIAM Journal on Numerical Analysis 1998,35(2):600–616. 10.1137/S0036142995288920
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Boulbrachene, M., Saadi, S. Maximum norm analysis of an overlapping nonmatching grids method for the obstacle problem. Adv Differ Equ 2006, 085807 (2006). https://doi.org/10.1155/ADE/2006/85807
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/ADE/2006/85807