An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type

Kassem Mustapha, Hermann Brunner, Hussein Mustapha, D Schoetzau

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28 Citations (Scopus)
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Abstract

We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an $hp$-version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal $hp$-version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near $t=0$ caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the $h$-version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems.
Original languageEnglish
Pages (from-to)1369-1396
Number of pages28
JournalSIAM Journal on Numerical Analysis
Volume49
Issue number4
DOIs
Publication statusPublished - 2011

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Keywords

  • hp-version
  • discontinuous galerkin
  • integro-differential equation
  • parabolic type

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