Divergence-Conforming Discontinuous Galerkin Methods and $C^0$ Interior Penalty Methods

Handle URI:
http://hdl.handle.net/10754/598001
Title:
Divergence-Conforming Discontinuous Galerkin Methods and $C^0$ Interior Penalty Methods
Authors:
Kanschat, Guido; Sharma, Natasha
Abstract:
© 2014 Society for Industrial and Applied Mathematics. In this paper, we show that recently developed divergence-conforming methods for the Stokes problem have discrete stream functions. These stream functions in turn solve a continuous interior penalty problem for biharmonic equations. The equivalence is established for the most common methods in two dimensions based on interior penalty terms. Then, extensions of the concept to discontinuous Galerkin methods defined through lifting operators, for different weak formulations of the Stokes problem, and to three dimensions are discussed. Application of the equivalence result yields an optimal error estimate for the Stokes velocity without involving the pressure. Conversely, combined with a recent multigrid method for Stokes flow, we obtain a simple and uniform preconditioner for harmonic problems with simply supported and clamped boundary.
Citation:
Kanschat G, Sharma N (2014) Divergence-Conforming Discontinuous Galerkin Methods and $C^0$ Interior Penalty Methods. SIAM J Numer Anal 52: 1822–1842. Available: http://dx.doi.org/10.1137/120902975.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Numerical Analysis
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
Jan-2014
DOI:
10.1137/120902975
Type:
Article
ISSN:
0036-1429; 1095-7170
Sponsors:
The first author was supported in part by the National Science Foundation through grant DMS-0810387 and by the King Abdullah University of Science and Technology (KAUST) through award KUS-C1-016-04. Part of this research was conceived and prepared when the author visited the Institute for Mathematics and Its Applications in Minneapolis.
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Full metadata record

DC FieldValue Language
dc.contributor.authorKanschat, Guidoen
dc.contributor.authorSharma, Natashaen
dc.date.accessioned2016-02-25T13:10:43Zen
dc.date.available2016-02-25T13:10:43Zen
dc.date.issued2014-01en
dc.identifier.citationKanschat G, Sharma N (2014) Divergence-Conforming Discontinuous Galerkin Methods and $C^0$ Interior Penalty Methods. SIAM J Numer Anal 52: 1822–1842. Available: http://dx.doi.org/10.1137/120902975.en
dc.identifier.issn0036-1429en
dc.identifier.issn1095-7170en
dc.identifier.doi10.1137/120902975en
dc.identifier.urihttp://hdl.handle.net/10754/598001en
dc.description.abstract© 2014 Society for Industrial and Applied Mathematics. In this paper, we show that recently developed divergence-conforming methods for the Stokes problem have discrete stream functions. These stream functions in turn solve a continuous interior penalty problem for biharmonic equations. The equivalence is established for the most common methods in two dimensions based on interior penalty terms. Then, extensions of the concept to discontinuous Galerkin methods defined through lifting operators, for different weak formulations of the Stokes problem, and to three dimensions are discussed. Application of the equivalence result yields an optimal error estimate for the Stokes velocity without involving the pressure. Conversely, combined with a recent multigrid method for Stokes flow, we obtain a simple and uniform preconditioner for harmonic problems with simply supported and clamped boundary.en
dc.description.sponsorshipThe first author was supported in part by the National Science Foundation through grant DMS-0810387 and by the King Abdullah University of Science and Technology (KAUST) through award KUS-C1-016-04. Part of this research was conceived and prepared when the author visited the Institute for Mathematics and Its Applications in Minneapolis.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectBiharmonicen
dc.subjectDivergence-free solutionsen
dc.subjectError estimatesen
dc.subjectFinite element cochain complexen
dc.subjectInterior penalty methodsen
dc.subjectPreconditioningen
dc.subjectStokes equationsen
dc.titleDivergence-Conforming Discontinuous Galerkin Methods and $C^0$ Interior Penalty Methodsen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Numerical Analysisen
dc.contributor.institutionUniversitat Heidelberg, Heidelberg, Germanyen
kaust.grant.numberKUS-C1-016-04en
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