A "natural" norm for the method of characteristics using discontinuous finite elements: 2D and 3D case

We consider the numerical approximation of a first order stationary hyperbo lic equation by the method of characteristics with pseudo time step k using discontinuous finite elements on a mesh tau(h). For this method, we exhibi t a "natural" norm parallel to parallel to(h,k) for which we show that the discrete variational problem P-h(k) is well posed and we obtain an error es timate. We show that when k goes to zero problem (P-h(k)) (resp. the parall el to parallel to(h,k) norm) has as a limit problem (P-h) (resp. the parall el to parallel to(h) norm) associated to the Galerkin discontinuous method. This extends to two and three space dimension our previous results obtaine d in one space dimension. AMS Subject Classification. 65N30, 65N25.