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hp-FEM for Hyperbolic Problems

by P. Houston and Ch. Schwab and E. Süli

(Report number 1999-14)

Abstract
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 mixed 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.

Keywords: hp finite element methods, hyperbolic problems, nonnegative characteristic form, a priori error analysis, a posteriori error analysis, adaptivity

@Techreport{HSS99_78,
  author = {P. Houston and Ch. Schwab and E. S\"uli},
  title = {hp-FEM for Hyperbolic Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1999-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1999/1999-14.pdf },
  year = {1999}
}

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