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

Guido Kanschat, Natasha Sharma

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)1822-1842
Number of pages21
JournalSIAM Journal on Numerical Analysis
Volume52
Issue number4
DOIs
StatePublished - Jan 2014
Externally publishedYes

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