# Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Yalchin Efendiev; Juan Galvis; Raytcho Lazarov; Joerg Willems

- Volume: 46, Issue: 5, page 1175-1199
- ISSN: 0764-583X

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topEfendiev, Yalchin, et al. "Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.5 (2012): 1175-1199. <http://eudml.org/doc/273336>.

@article{Efendiev2012,

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.},

author = {Efendiev, Yalchin, Galvis, Juan, Lazarov, Raytcho, Willems, Joerg},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {domain decomposition; robust additive Schwarz preconditioner; spectral coarse spaces; high contrast; Brinkman’s problem; multiscale problems; Brinkman's problem; porous media; elliptic (pressure) equation; finite element; second-order elliptic problem; Stokes equation; numerical experiments},

language = {eng},

number = {5},

pages = {1175-1199},

publisher = {EDP-Sciences},

title = {Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms},

url = {http://eudml.org/doc/273336},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Efendiev, Yalchin

AU - Galvis, Juan

AU - Lazarov, Raytcho

AU - Willems, Joerg

TI - Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2012

PB - EDP-Sciences

VL - 46

IS - 5

SP - 1175

EP - 1199

AB - 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.

LA - eng

KW - domain decomposition; robust additive Schwarz preconditioner; spectral coarse spaces; high contrast; Brinkman’s problem; multiscale problems; Brinkman's problem; porous media; elliptic (pressure) equation; finite element; second-order elliptic problem; Stokes equation; numerical experiments

UR - http://eudml.org/doc/273336

ER -

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