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Conditions for superconvergence of HDG methods for second-order elliptic problems


Authors: Bernardo Cockburn, Weifeng Qiu and Ke Shi
Journal: Math. Comp. 81 (2012), 1327-1353
MSC (2010): Primary 35L65, 65M60, 65N30
DOI: https://doi.org/10.1090/S0025-5718-2011-02550-0
Published electronically: October 13, 2011
MathSciNet review: 2904581
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Abstract: We provide a projection-based analysis of a large class of finite element methods for second order elliptic problems. It includes the hybridized version of the main mixed and hybridizable discontinuous Galerkin methods. The main feature of this unifying approach is that it reduces the main difficulty of the analysis to the verification of some properties of an auxiliary, locally defined projection and of the local spaces defining the methods. Sufficient conditions for the optimal convergence of the approximate flux and the superconvergence of an element-by-element postprocessing of the scalar variable are obtained. New mixed and hybridizable discontinuous Galerkin methods with these properties are devised which are defined on squares, cubes and prisms.


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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Weifeng Qiu
Affiliation: Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
Email: qiuxa001@ima.umn.edu

Ke Shi
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: shixx075@math.umn.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02550-0
Received by editor(s): October 12, 2010
Received by editor(s) in revised form: March 10, 2011
Published electronically: October 13, 2011
Additional Notes: Supported in part by the National Science Foundation (Grant DMS-0712955) and by the University of Minnesota Supercomputing Institute.
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society