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Superconvergent discontinuous Galerkin methods for second-order elliptic problems


Authors: Bernardo Cockburn, Johnny Guzmán and Haiying Wang
Journal: Math. Comp. 78 (2009), 1-24
MSC (2000): Primary 65M60, 65N30, 35L65
DOI: https://doi.org/10.1090/S0025-5718-08-02146-7
Published electronically: May 19, 2008
MathSciNet review: 2448694
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Abstract: We identify discontinuous Galerkin methods for second-order elliptic problems in several space dimensions having superconvergence properties similar to those of the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These methods use polynomials of degree $ k\ge0$ for both the potential as well as the flux. We show that the approximate flux converges in $ L^2$ with the optimal order of $ k+1$, and that the approximate potential and its numerical trace superconverge, in $ L^2$-like norms, to suitably chosen projections of the potential, with order $ k+2$. We also apply element-by-element postprocessing of the approximate solution to obtain new approximations of the flux and the potential. The new approximate flux is proven to have normal components continuous across inter-element boundaries, to converge in $ L^2$ with order $ k+1$, and to have a divergence converging in $ L^2$ also with order $ k+1$. The new approximate potential is proven to converge with order $ k+2$ in $ L^2$. Numerical experiments validating these theoretical results are presented.


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Additional Information

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

Johnny Guzmán
Affiliation: School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, Minnesota 55455
Email: guzman033@math.umn.edu

Haiying Wang
Affiliation: Reservoir Engineering Research Institute, 385 Sherman Avenue, Suite 5, Palo Alto, California 94306
Email: hywang@rerinst.org

DOI: https://doi.org/10.1090/S0025-5718-08-02146-7
Keywords: Finite element methods, mixed methods, discontinuous Galerkin methods, superconvergence, postprocessing
Received by editor(s): October 9, 2007
Received by editor(s) in revised form: January 31, 2008
Published electronically: May 19, 2008
Additional Notes: B. Cockburn was supported in part by the National Science Foundation (Grant DMS-0712955) and by the University of Minnesota Supercomputing Institute
J. Guzmán was supported by a National Science Foundation Mathematical Science Postdoctoral Research Fellowship (DMS-0503050)
Article copyright: © Copyright 2008 American Mathematical Society

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