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An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems

Authors: Caroline Lasser and Andrea Toselli
Journal: Math. Comp. 72 (2003), 1215-1238
MSC (2000): Primary 65F10, 65N22, 65N30, 65N55
Published electronically: January 8, 2003
MathSciNet review: 1972733
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Abstract: We consider a scalar advection-diffusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, nonsymmetric linear system, we propose and study an additive, two-level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions.

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Additional Information

Caroline Lasser
Affiliation: Center for Mathematical Sciences, Technische Universität München, D-85748 Garching bei München, Germany

Andrea Toselli
Affiliation: Seminar for Applied Mathematics, ETH Zürich, Rämisstr. 101, CH-8092 Zürich, Switzerland

Keywords: Advection-diffusion problem, domain decomposition, discontinuous Galerkin approximation, preconditioning
Received by editor(s): November 14, 2000
Received by editor(s) in revised form: April 3, 2001
Published electronically: January 8, 2003
Additional Notes: The work of the first author was supported by the Studienstiftung des Deutschen Volkes while she was visiting the Courant Institute of Mathematical Sciences
Most of this work was carried out while the second author was affiliated to the Courant Institute of Mathematical Sciences, New York, and supported in part by the Applied Mathematical Sciences Program of the U.S. Department of Energy under Contract DEFGO288ER25053.
Article copyright: © Copyright 2003 American Mathematical Society