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An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. (English) Zbl 0618.65105
We prove $$L_ p$$ 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 $$L_ 2$$ estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. $$L_ p$$ estimates for $$p\neq 2$$ 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.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35L50 Initial-boundary value problems for first-order hyperbolic systems
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