\(hp\)-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].

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 |