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Discontinuous hp-Finite Element Methods for Advection-Diffusion Problems

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

(Report number 2000-07)

Abstract
We consider the hp-version of the discontinuous Galerkin finite element method for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, first-order hyperbolic equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by 1/2 a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.

Keywords: hp-finite element methods, discontinuous Galerkin methods, PDEs with nonnegative characteristic form

@Techreport{HSS00_266,
  author = {P. Houston and Ch. Schwab and E. S\"uli},
  title = {Discontinuous hp-Finite Element Methods for Advection-Diffusion Problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2000-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2000/2000-07.pdf },
  year = {2000}
}

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