On fully discrete Galerkin approximations for partial integrodifferential equations of parabolic type
Author:
Nai Ying Zhang
Journal:
Math. Comp. 60 (1993), 133166
MSC:
Primary 65M60; Secondary 35K05, 65M15
MathSciNet review:
1149295
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Additional Information
Abstract: The subject of this work is the application of fully discrete Galerkin finite element methods to initialboundary value problems for linear partial integrodifferential equations of parabolic type. We investigate numerical schemes based on the Padé discretization with respect to time and associated with certain quadrature formulas to approximate the integral term. A preliminary error estimate is established, which contains a term related to the quadrature rule to be specified. In particular, we consider quadrature rules with sparse quadrature points so as to limit the storage requirements, without sacrificing the order of overall convergence. For the backward Euler scheme, the CrankNicolson scheme, and a thirdorder (1,2) Padétype scheme, the specific quadrature rules analyzed are based on the rectangular, the trapezoidal, and Simpson's rule. For all the schemes studied, optimalorder error estimates are obtained in the case that the solution of the problem is smooth enough. Since this is important for our error analysis, we also discuss the regularity of the exact solutions of our equations. Highorder regularity results with respect to both space and time are given for the solution of problems with smooth enough data.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199311492954
PII:
S 00255718(1993)11492954
Keywords:
Initialboundary value problem,
parabolic,
partial,
integrodifferential equation,
regularity,
finite element method,
fully discrete,
Galerkin approximation,
stability,
error estimate,
quadrature
Article copyright:
© Copyright 1993
American Mathematical Society
