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 $hp$-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.