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$ hp$-Optimal discontinuous Galerkin methods for linear elliptic problems


Authors: Benjamin Stamm and Thomas P. Wihler
Journal: Math. Comp. 79 (2010), 2117-2133
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-10-02335-5
Published electronically: April 9, 2010
MathSciNet review: 2684358
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Abstract: The aim of this paper is to present and analyze a class of $ hp$-version discontinuous Galerkin (DG) discretizations for the numerical approximation of linear elliptic problems. This class includes a number of well-known DG formulations. We will show that the methods are stable provided that the stability parameters are suitably chosen. Furthermore, on (possibly irregular) quadrilateral meshes, we shall prove that the schemes converge all optimally in the energy norm with respect to both the local element sizes and polynomial degrees provided that homogeneous boundary conditions are considered.


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Additional Information

Benjamin Stamm
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Box F, Providence, RI 02912
Email: Benjamin_Stamm@Brown.edu

Thomas P. Wihler
Affiliation: Mathematics Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Email: wihler@math.unibe.ch

DOI: https://doi.org/10.1090/S0025-5718-10-02335-5
Keywords: $hp$-methods, discontinuous Galerkin methods, optimal error estimates.
Received by editor(s): October 26, 2007
Received by editor(s) in revised form: June 20, 2009
Published electronically: April 9, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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