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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Uniform-in-time superconvergence of HDG methods for the heat equation
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by Brandon Chabaud and Bernardo Cockburn PDF
Math. Comp. 81 (2012), 107-129 Request permission

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 $k\ge 0$ 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 $k\ge 1$ uniformly in time. As a consequence, an element-by-element postprocessing converges with the rate $\sqrt {\log (T/h^2)} h^{k+2}$ for $k\ge 1$ 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
  • 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.
  • © Copyright 2011 American Mathematical Society
  • Journal: Math. Comp. 81 (2012), 107-129
  • MSC (2010): Primary 65M60, 35K05
  • DOI: https://doi.org/10.1090/S0025-5718-2011-02525-1
  • MathSciNet review: 2833489