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Time Discretization of Parabolic Problems by the hp-Version of the Discontinuous Galerkin Finite Element Method

by D. Schötzau and Ch. Schwab

(Report number 1999-04)

Abstract
The Discontinuous Galerkin Finite Element Method (DGFEM) for the time discretization of parabolic problems is analyzed in a hp-version context. Error bounds which are explicit in the time step as well as the approximation order are derived and it is shown that the hp-DGFEM gives spectral convergence in problems with smooth time dependence. In conjunction with geometric time partitions it is proved that the hp-DGFEM results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms. For the h-version DGFEM algebraically graded time partitions are determined that give the optimal algebraic convergence rates. A fully discrete hp scheme is discussed exemplarily for the heat equation. The use of certain mesh-design principles for spatial discretizations yields exponential rates of convergence in time and space. Numerical examples confirm the theoretical results.

Keywords: Abstract Parabolic Problems, Discontinuous Galerkin Methods, hp-Version of the Finite Element Method

@Techreport{SS99_239,
  author = {D. Sch\"otzau and Ch. Schwab},
  title = {Time Discretization of Parabolic Problems by the hp-Version of the Discontinuous Galerkin Finite Element Method},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1999-04},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1999/1999-04.pdf },
  year = {1999}
}

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