Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method

Authors:
Stig Larsson, Vidar Thomée and Lars B. Wahlbin

Journal:
Math. Comp. **67** (1998), 45-71

MSC (1991):
Primary 65M60, 65R20, 45L10

DOI:
https://doi.org/10.1090/S0025-5718-98-00883-7

MathSciNet review:
1432129

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Abstract: The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, can then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretization in space is also studied.

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Additional Information

**Stig Larsson**

Affiliation:
Department of Mathematics, Chalmers University of Technology and Göteborg University, S–412 96 Göteborg, Sweden

Email:
stig@math.chalmers.se

**Vidar Thomée**

Affiliation:
Department of Mathematics, Chalmers University of Technology and Göteborg University, S–412 96 Göteborg, Sweden

Email:
thomee@math.chalmers.se

**Lars B. Wahlbin**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Email:
wahlbin@math.cornell.edu

DOI:
https://doi.org/10.1090/S0025-5718-98-00883-7

Keywords:
Parabolic integro-differential equation,
weakly singular kernel,
discontinuous Galerkin,
variable time step,
finite element,
error estimate,
Gronwall lemma

Received by editor(s):
October 10, 1995

Received by editor(s) in revised form:
August 5, 1996

Additional Notes:
The first two authors were partly supported by the Swedish Research Council for Engineering Sciences (TFR). The third author thanks the National Science Foundation, USA, for financial support and also Chalmers University of Technology and Göteborg University for their hospitality during the Spring of 1995.

Article copyright:
© Copyright 1998
American Mathematical Society