Conditions for superconvergence of HDG methods for second-order elliptic problems
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- by Bernardo Cockburn, Weifeng Qiu and Ke Shi PDF
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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.References
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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
- MR Author ID: 845089
- Email: qiuxa001@ima.umn.edu
- Ke Shi
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 904733
- Email: shixx075@math.umn.edu
- 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.
- © Copyright 2011 American Mathematical Society
- Journal: Math. Comp. 81 (2012), 1327-1353
- MSC (2010): Primary 35L65, 65M60, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2011-02550-0
- MathSciNet review: 2904581