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Constructively well-posed approximation methods with unity inf-sup and continuity constants for partial differential equations


Authors: Tan Bui-Thanh, Leszek Demkowicz and Omar Ghattas
Journal: Math. Comp. 82 (2013), 1923-1952
MSC (2010): Primary 65N30; Secondary 65N12, 65N15, 65N22
Published electronically: April 23, 2013
MathSciNet review: 3073186
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Abstract: Starting from the generalized Lax-Milgram theorem and from the fact that the approximation error is minimized when the continuity and inf-sup constants are unity, we develop a theory that provably delivers well-posed approximation methods with unity continuity and inf-sup constants for numerical solution of linear partial differential equations. We demonstrate our single-framework theory on scalar hyperbolic equations to constructively derive two different $ hp$ finite element methods. The first one coincides with a least squares discontinuous Galerkin method, and the other appears to be new. Both methods are proven to be trivially well-posed, with optimal $ hp$-convergence rates. The numerical results show that our new discontinuous finite element method, namely a discontinuous Petrov-Galerkin method, is more accurate, has optimal convergence rate, and does not seem to have nonphysical diffusion compared to the upwind discontinuous Galerkin method.


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Additional Information

Tan Bui-Thanh
Affiliation: Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, Texas 78712
Email: tanbui@ices.utexas.edu

Leszek Demkowicz
Affiliation: Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, Texas 78712
Email: leszek@ices.utexas.edu

Omar Ghattas
Affiliation: Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, Texas 78712
Email: omar@ices.utexas.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02697-X
Keywords: Inf-sup condition, inf-sup constant, generalized Lax-Milgram theorem, discontinuous Galerkin method, discontinuous Petrov-Galerkin method, consistency, finite element method, stability, convergence, well-posedness hyperbolic partial differential equations
Received by editor(s): April 18, 2011
Received by editor(s) in revised form: October 31, 2011
Published electronically: April 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society