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Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Published online by Cambridge University Press:  22 February 2012

Yalchin Efendiev
Affiliation:
Dept. Mathematics, Texas A&M University, College Station, Texas 77843, USA. efendiev@math.tamu.edu; jugal@math.tamu.edu; lazarov@math.tamu.edu
Juan Galvis
Affiliation:
Dept. Mathematics, Texas A&M University, College Station, Texas 77843, USA. efendiev@math.tamu.edu; jugal@math.tamu.edu; lazarov@math.tamu.edu
Raytcho Lazarov
Affiliation:
Dept. Mathematics, Texas A&M University, College Station, Texas 77843, USA. efendiev@math.tamu.edu; jugal@math.tamu.edu; lazarov@math.tamu.edu
Joerg Willems
Affiliation:
Radon Institute for Computational and Applied Mathematics (RICAM), Altenberger Strasse 69, 4040 Linz, Austria; joerg.willems@ricam.oeaw.ac.at
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Abstract

An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Galvis and Efendiev [Multiscale Model. Simul. 8 (2010) 1461–1483] developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman’s problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2012

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