Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Yalchin Efendiev; Juan Galvis; Raytcho Lazarov; Joerg Willems

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

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

Abstract

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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.

How to cite

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Efendiev, 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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