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$$hp$$-finite element methods for hyperbolic problems. (English) Zbl 0959.65127
Whiteman, J. R. (ed.), The mathematics of finite elements and applications X, MAFELAP 1999. Proceedings of the 10th conference, Brunel Univ., Uxbridge, Middlesex, GB, June 22-25, 1999. Amsterdam: Elsevier. 143-162 (2000).
Summary: This paper is devoted to the a priori and a posteriori error analysis of the $$hp$$-version of the discontinuous Galerkin finite element method for partial differential equations of hyperbolic and nearly-hyperbolic character. We consider second-order partial differential equations with nonnegative characteristic form, a large class of equations which includes convection-dominated diffusion problems, degenerate elliptic equations and second-order problems of mxied elliptic-hyperbolic-parabolic type.
An a priori error bound is derived for the method in the so-called DG-norm which is optimal in terms of the mesh size $$h$$; the error bound is either 1 degree or 1/2 degree below optimal in terms of the polynomial degree $$p$$, depending on whether the problem is convection-dominated, or diffusion-dominated, respectively.
In the case of a first-order hyperbolic equation the error bound is $$hp$$-optimal in the DG-norm. For first-order hyperbolic problems, we also discuss the a posteriori error analysis of the method and implement the resulting bounds into an $$hp$$-adaptive algorithm. The theoretical findings are illustrated by numerical experiments.
For the entire collection see [Zbl 0942.00044].

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35L20 Initial-boundary value problems for second-order hyperbolic equations 35M10 PDEs of mixed type 65N15 Error bounds for boundary value problems involving PDEs 35L50 Initial-boundary value problems for first-order hyperbolic systems 35J25 Boundary value problems for second-order elliptic equations