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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
Published electronically: July 14, 2011
MathSciNet review: 2833489
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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 $ k$, the approximation to the gradient converges with the rate $ h^{k+1}$ for $ k\ge0$ and the $ L^2$-projection of the error into a space of lower polynomial degree superconverges with the rate $ \sqrt{\log (T/h^2)} h^{k+2}$ for $ k\ge1$ uniformly in time. As a consequence, an element-by-element postprocessing converges with the rate $ \sqrt{\log (T/h^2)} h^{k+2}$ for $ k\ge1$ 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: http://dx.doi.org/10.1090/S0025-5718-2011-02525-1
PII: S 0025-5718(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