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

Abstract

We prove Lp 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 L2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. L estimates for p =* 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.

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