Volume 48, pp. 156-182, 2018.

Multiscale coarse spaces for overlapping Schwarz methods based on the ACMS space in 2D

Alexander Heinlein, Axel Klawonn, Jascha Knepper, and Oliver Rheinbach

Abstract

Two-level overlapping Schwarz domain decomposition methods for second-order elliptic problems in two dimensions are proposed using coarse spaces constructed from the Approximate Component Mode Synthesis (ACMS) multiscale discretization approach. These coarse spaces are based on eigenvalue problems using Schur complements on subdomain edges. It is then shown that the convergence of the resulting preconditioned Krylov method can be controlled by a user-specified tolerance and thus can be made independent of heterogeneities in the coefficient of the partial differential equation. The relations of this new approach to other known adaptive coarse space approaches for overlapping Schwarz methods are also discussed. Compared to one of the competing adaptive approaches, the new coarse space can be significantly smaller. Compared to other competing approaches, the eigenvalue problems are significantly cheaper to solve, i.e., the dimension of the eigenvalue problems is minimal among the competing adaptive approaches under consideration. Our local eigenvalue problems can be solved using one iteration of LobPCG for essentially the same cost as a Cholesky-decomposition of a Schur complement on a subdomain edge.

Full Text (PDF) [459 KB]

Key words

domain decomposition, multiscale, approximate component mode synthesis (ACMS), scientific software, parallel, overlapping Schwarz

AMS subject classifications

65F08, 65F10, 65N55, 68W10, 74S30

Links to the cited ETNA articles

[24]Vol. 41 (2014), pp. 109-132 Ulrich Hetmaniuk and Axel Klawonn: Error estimates for a two-dimensional special finite element method based on component mode synthesis
[31]Vol. 45 (2016), pp. 75-106 Axel Klawonn, Patrick Radtke, and Oliver Rheinbach: A comparison of adaptive coarse spaces for iterative substructuring in two dimensions
[37]Vol. 46 (2017), pp. 273-336 Clemens Pechstein and Clark R. Dohrmann: A unified framework for adaptive BDDC

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