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.


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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