Divergence-Conforming Discontinuous Galerkin Methods and Interior Penalty Methods

Type
Article

Authors
Kanschat, Guido
Sharma, Natasha

KAUST Grant Number
KUS-C1-016-04

Date
2014-01

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 Interior Penalty Methods. SIAM J Numer Anal 52: 1822–1842. Available: http://dx.doi.org/10.1137/120902975.

Acknowledgements
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.

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Numerical Analysis

DOI
10.1137/120902975

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