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Mathematics of Computation

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A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems


Authors: Bernardo Cockburn, Bo Dong and Johnny Guzmán
Journal: Math. Comp. 77 (2008), 1887-1916
MSC (2000): Primary 65M60, 65N30, 35L65
Published electronically: May 6, 2008
MathSciNet review: 2429868
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Abstract: We identify and study an LDG-hybridizable Galerkin method, which is not an LDG method, for second-order elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuous Galerkin methods using polynomials of degree $ k\ge0$ for both the potential as well as the flux, the order of convergence in $ L^2$ of both unknowns is $ k+1$. Moreover, both the approximate potential as well as its numerical trace superconverge in $ L^2$-like norms, to suitably chosen projections of the potential, with order $ k+2$. This allows the application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential converging with order $ k+2$ in $ L^2$. The method can be thought to be in between the hybridized version of the Raviart-Thomas and that of the Brezzi-Douglas-Marini mixed methods.


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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Bo Dong
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: bdong@dam.brown.edu

Johnny Guzmán
Affiliation: School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455
Email: guzma033@umn.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02123-6
Keywords: Discontinuous Galerkin methods, hybridization, superconvergence, second-order elliptic problems
Received by editor(s): November 1, 2006
Received by editor(s) in revised form: September 6, 2007
Published electronically: May 6, 2008
Additional Notes: The first author was supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute.
The third author was supported by an NSF Mathematical Science Postdoctoral Research Fellowship (DMS-0503050)
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.