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 orderstationary 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 weobtain 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 discontinuousmethod. This extends to two and three space dimension our previous results obtained in one space dimension.