Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz Methods

Journal article (2022)

Authors

A. Heinlein Delft Institute of Applied Mathematics , Numerical Analysis - EEMCS
A. Klawonn University of Cologne
Jascha Knepper University of Cologne
O. Rheinbach University of Technology Bergakademie Freiberg
Olof B. Widlund Courant Institute of Mathematical Sciences
Research Group
Numerical Analysis (EEMCS) (TU Delft)
Copyright: © 2022 A. Heinlein, Axel Klawonn, Jascha Knepper, Oliver Rheinbach, Olof B. Widlund
DOI: https://doi.org/10.1137/20M1364540
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Published Date

2022

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Numerical Analysis

Abstract

A new reduced-dimension adaptive generalized Dryja-Smith-Widlund (GDSW) overlapping Schwarz method for linear second-order elliptic problems in three dimensions is introduced. It is robust with respect to large contrasts of the coefficients of the partial differential equations. The condition number bound of the new method is shown to be independent of the coefficient contrast and only dependent on a user-prescribed tolerance. The interface of the nonoverlapping domain decomposition is partitioned into nonoverlapping patches. The new coarse space is obtained by selecting a few eigenvectors of certain local eigenproblems which are defined on these patches. These eigenmodes are energy-minimally extended to the interior of the nonoverlapping subdomains and added to the coarse space. By using a new interface decomposition, the reduced-dimension adaptive GDSW overlapping Schwarz method usually has a smaller coarse space than existing GDSW and adaptive GDSW domain decomposition methods. A robust condition number estimate is proven for the new reduced-dimension adaptive GDSW method which is also valid for existing adaptive GDSW methods. Numerical results for the equations of isotropic linear elasticity in three dimensions confirming the theoretical findings are presented.

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