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

Keywords

Domain decompositionrobust additive Schwarz preconditionerspectral coarse spaceshigh contrastBrinkman’s problemmultiscale problems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1175 - 1199
Copyright
© EDP Sciences, SMAI, 2012

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