Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients

Handle URI:
http://hdl.handle.net/10754/594284
Title:
Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients
Authors:
Ayuso Dios, Blanca; Holst, Michael; Zhu, Yunrong; Zikatanov, Ludmil
Abstract:
We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. © 2013 American Mathematical Society.
KAUST Department:
Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Ayuso de Dios B, Holst M, Zhu Y, Zikatanov L (2013) Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients. Math Comp 83: 1083–1120. Available: http://dx.doi.org/10.1090/s0025-5718-2013-02760-3.
Publisher:
American Mathematical Society (AMS)
Journal:
Mathematics of Computation
Issue Date:
30-Oct-2013
DOI:
10.1090/s0025-5718-2013-02760-3
Type:
Article
ISSN:
0025-5718; 1088-6842
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorAyuso Dios, Blancaen
dc.contributor.authorHolst, Michaelen
dc.contributor.authorZhu, Yunrongen
dc.contributor.authorZikatanov, Ludmilen
dc.date.accessioned2016-01-19T14:45:10Zen
dc.date.available2016-01-19T14:45:10Zen
dc.date.issued2013-10-30en
dc.identifier.citationAyuso de Dios B, Holst M, Zhu Y, Zikatanov L (2013) Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients. Math Comp 83: 1083–1120. Available: http://dx.doi.org/10.1090/s0025-5718-2013-02760-3.en
dc.identifier.issn0025-5718en
dc.identifier.issn1088-6842en
dc.identifier.doi10.1090/s0025-5718-2013-02760-3en
dc.identifier.urihttp://hdl.handle.net/10754/594284en
dc.description.abstractWe introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. © 2013 American Mathematical Society.en
dc.publisherAmerican Mathematical Society (AMS)en
dc.subjectCrouzeix-Raviart finite elementsen
dc.subjectDiscontinuous Galerkin methodsen
dc.subjectMultilevel preconditioneren
dc.subjectSpace decompositionen
dc.titleMultilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficientsen
dc.typeArticleen
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)en
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalMathematics of Computationen
dc.contributor.institutionCentre de Recerca Matematica, Campus de Bellaterra, Bellaterra, 08193, Spainen
dc.contributor.institutionDepartment of Mathematics, University of California, San Diego, La Jolla, CA 92093, United Statesen
dc.contributor.institutionDepartment of Mathematics, Idaho State University, Pocatello, ID 83209-8085, United Statesen
dc.contributor.institutionDepartment of Mathematics, Pennsylvania State University, University Park, PA 16802, United Statesen
kaust.authorAyuso Dios, Blancaen
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