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

K. Mustapha*, Hermann BRUNNER, H. Mustapha, D. Schotzau

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

37 Citations (Scopus)

Abstract

We study the numerical solution of a class of pa rabolic 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

Scopus Subject Areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Exponential convergence
  • Finite element method
  • Fully discrete scheme
  • Hp-version DG time-stepping
  • Parabolic volterra integro-differential equation
  • Weakly singular kernel

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