An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation

Abstract

We prove L p {L_p} stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain L 2 {L_2} estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. L p {L_p} estimates for p ≠ 2 p \ne 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.