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Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization: The many-subdomain case

Authors: Martin J. Gander and Soheil Hajian
Journal: Math. Comp. 87 (2018), 1635-1657
MSC (2010): Primary 65N22, 65F10, 65F08, 65N55, 65H10
Published electronically: September 29, 2017
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Abstract: Schwarz methods are attractive parallel solution techniques for solving large-scale linear systems obtained from discretizations of partial differential equations (PDEs). Due to the iterative nature of Schwarz methods, convergence rates are an important criterion to quantify their performance. Optimized Schwarz methods (OSM) form a class of Schwarz methods that are designed to achieve faster convergence rates by employing optimized transmission conditions between subdomains. It has been shown recently that for a two-subdomain case, OSM is a natural solver for hybridizable discontinuous Galerkin (HDG) discretizations of elliptic PDEs. In this paper, we generalize the preceding result to the many-subdomain case and obtain sharp convergence rates with respect to the mesh size and polynomial degree, the subdomain diameter, and the zeroth-order term of the underlying PDE, which allows us for the first time to give precise convergence estimates for OSM used to solve parabolic problems by implicit time stepping. We illustrate our theoretical results with numerical experiments.

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Additional Information

Martin J. Gander
Affiliation: Section de mathématiques, Université de Genève, Geneva, Switzerland

Soheil Hajian
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

Keywords: Additive Schwarz, optimized Schwarz, discontinuous Galerkin methods, scalability, parabolic problems
Received by editor(s): March 15, 2016
Received by editor(s) in revised form: February 6, 2017
Published electronically: September 29, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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