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

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 hpconvergence 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.

DOI: 10.1090/S0025-5718-2013-02697-X

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Cite this paper

@article{BuiThanh2013ConstructivelyWA, title={Constructively well-posed approximation methods with unity inf-sup and continuity constants for partial differential equations}, author={Tan Bui-Thanh and Leszek F. Demkowicz and Omar Ghattas}, journal={Math. Comput.}, year={2013}, volume={82}, pages={1923-1952} }