A “Natural” Norm for the Method of Characteristics Using Discontinuous Finite Elements : 2D and 3D case

Abstract

We consider the numerical approximation of a first order stationary hyperbolic equation by the method of characteristics with pseudo time step k using discontinuous finite elements on a mesh ${\cal T}_h$ . For this method, we exhibit a “natural” norm || || h,k for which we show that the discrete variational problem $P_h^k$ is well posed and we obtain an error estimate. We show that when k goes to zero problem $(P_h^k)$ (resp. the || || h,k norm) has as a limit problem ( P h ) (resp. the || || h norm) associated to the Galerkin discontinuous method. This extends to two and three space dimension our previous results obtained in one space dimension.