Optimized Schwarz and 2-Lagrange Multiplier Methods for Multiscale Elliptic PDEs

Sebastien Loisel, Trung Hieu Nguyen, Robert Scheichl

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6 Citations (Scopus)
52 Downloads (Pure)

Abstract

In this article, we formulate and analyze a two-level preconditioner for optimized Schwarz and 2-Lagrange multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of low-frequency modes of approximate subdomain Dirichlet-to-Neumann maps. Under a suitable change of basis, the preconditioner is a $2 \times 2$ block upper triangular matrix with the identity matrix in the upper-left block. We show that the spectrum of the preconditioned system is included in the disk having center $z=1/2$ and radius $r=1/2 - \epsilon$, where $0 < \epsilon < 1/2$ is a parameter that we can choose. We further show that the GMRES algorithm applied to our heterogeneous system converges in $O(1/\epsilon)$ iterations (neglecting certain polylogarithmic terms). The number $\epsilon$ can be made arbitrarily large by automatically enriching the coarse space. Our theoretical results are confirmed by numerical experiments.
Original languageEnglish
Pages (from-to)A2896–A2923
Number of pages28
JournalSIAM Journal on Scientific Computing
Volume37
Issue number6
Early online date10 Dec 2015
DOIs
Publication statusPublished - 2015

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