Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Handle URI:
http://hdl.handle.net/10754/599521
Title:
Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms
Authors:
Efendiev, Yalchin; Galvis, Juan; Lazarov, Raytcho; Willems, Joerg
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. © EDP Sciences, SMAI, 2012.
Citation:
Efendiev Y, Galvis J, Lazarov R, Willems J (2012) Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM: Mathematical Modelling and Numerical Analysis 46: 1175–1199. Available: http://dx.doi.org/10.1051/m2an/2011073.
Publisher:
EDP Sciences
Journal:
ESAIM: Mathematical Modelling and Numerical Analysis
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
22-Feb-2012
DOI:
10.1051/m2an/2011073
Type:
Article
ISSN:
0764-583X; 1290-3841
Sponsors:
The research of Y. Efendiev, J. Galvis and R. Lazarov was supported in parts by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). The research of R. Lazarov and J. Willems was supported in parts by NSF Grant DMS-1016525.
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Full metadata record

DC FieldValue Language
dc.contributor.authorEfendiev, Yalchinen
dc.contributor.authorGalvis, Juanen
dc.contributor.authorLazarov, Raytchoen
dc.contributor.authorWillems, Joergen
dc.date.accessioned2016-02-28T05:52:41Zen
dc.date.available2016-02-28T05:52:41Zen
dc.date.issued2012-02-22en
dc.identifier.citationEfendiev Y, Galvis J, Lazarov R, Willems J (2012) Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM: Mathematical Modelling and Numerical Analysis 46: 1175–1199. Available: http://dx.doi.org/10.1051/m2an/2011073.en
dc.identifier.issn0764-583Xen
dc.identifier.issn1290-3841en
dc.identifier.doi10.1051/m2an/2011073en
dc.identifier.urihttp://hdl.handle.net/10754/599521en
dc.description.abstractAn 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. © EDP Sciences, SMAI, 2012.en
dc.description.sponsorshipThe research of Y. Efendiev, J. Galvis and R. Lazarov was supported in parts by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). The research of R. Lazarov and J. Willems was supported in parts by NSF Grant DMS-1016525.en
dc.publisherEDP Sciencesen
dc.subjectBrinkman's problemen
dc.subjectDomain decompositionen
dc.subjectHigh contrasten
dc.subjectMultiscale problemsen
dc.subjectRobust additive Schwarz preconditioneren
dc.subjectSpectral coarse spacesen
dc.titleRobust domain decomposition preconditioners for abstract symmetric positive definite bilinear formsen
dc.typeArticleen
dc.identifier.journalESAIM: Mathematical Modelling and Numerical Analysisen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionJohann Radon Institute for Computational and Applied Mathematics, Linz, Austriaen
kaust.grant.numberKUS-C1-016-04en
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