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Numerical homogenization of elliptic PDEs with similar coefficients

Journal article
Authors F. Hellman
Axel Målqvist
Published in Multiscale Modeling and Simulation
Volume 17
Issue 2
Pages 650-674
ISSN 1540-3459
Publication year 2019
Published at Department of Mathematical Sciences
Pages 650-674
Language en
Links dx.doi.org/10.1137/18M1189701
Keywords Elliptic PDEs, Finite element method, Numerical homogenization, Time-dependent PDEs
Subject categories Computational Mathematics

Abstract

We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per time step) and in sample based stochastic integration of outputs from an elliptic PDE (with one coefficient per sample member). We propose a parallelizable algorithm based on Petrov–Galerkin localized orthogonal decomposition that adaptively (using computable and theoretically derived error indicators) recomputes the local corrector problems only where it improves accuracy. The method is illustrated in detail by an example of a time-dependent two-pase Darcy flow problem in three dimensions. c 2019 Society for Industrial and Applied Mathematics

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