Uniform-in-time superconvergence of HDG methods for the heat equation

Authors:
Brandon Chabaud and Bernardo Cockburn

Journal:
Math. Comp. **81** (2012), 107-129

MSC (2010):
Primary 65M60, 35K05

DOI:
https://doi.org/10.1090/S0025-5718-2011-02525-1

Published electronically:
July 14, 2011

MathSciNet review:
2833489

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the superconvergence properties of the hybridizable discontinuous Galerkin method for second-order elliptic problems do hold uniformly in time for the semidiscretization by the same method of the heat equation provided the solution is smooth enough. Thus, if the approximations are piecewise polynomials of degree , the approximation to the gradient converges with the rate for and the -projection of the error into a space of lower polynomial degree *superconverges * with the rate for uniformly in time. As a consequence, an element-by-element postprocessing converges with the rate for also uniformly in time. Similar results are proven for the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods.

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Additional Information

**Brandon Chabaud**

Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802

Email:
chabaud@math.psu.edu

**Bernardo Cockburn**

Affiliation:
School of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, Minnesota 55455

Email:
cockburn@math.umn.edu

DOI:
https://doi.org/10.1090/S0025-5718-2011-02525-1

Keywords:
Discontinuous Galerkin methods,
hybridization,
superconvergence,
parabolic problems

Received by editor(s):
January 27, 2010

Received by editor(s) in revised form:
July 26, 2010

Published electronically:
July 14, 2011

Additional Notes:
Part of this work was done when the first author was at the School of Mathematics, University of Minnesota.

The second author was partially supported by the National Science Foundation (Grant DMS-0712955) and by the Minnesota Supercomputing Institute. Part of this work was done when this author was visiting the Research Institute for Mathematical Sciences, Kyoto University, Japan, during the Fall of 2009.

Article copyright:
© Copyright 2011
American Mathematical Society