Error estimates for semidiscrete finite element methods for parabolic integro-differential equations

Authors:
Vidar Thomée and Nai Ying Zhang

Journal:
Math. Comp. **53** (1989), 121-139

MSC:
Primary 65R20; Secondary 65M60

DOI:
https://doi.org/10.1090/S0025-5718-1989-0969493-9

MathSciNet review:
969493

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to attempt to carry over known results for spatially discrete finite element methods for linear parabolic equations to integro-differential equations of parabolic type with an integral kernel consisting of a partial differential operator of order . It is shown first that this is possible without restrictions when the exact solution is smooth. In the case of a homogeneous equation with nonsmooth initial data , optimal convergence for positive time is possible in general only if . This depends on the fact that the exact solution is then only in .

**[1]**J. H. Bramble, A. H. Schatz, V. Thomée & L. B. Wahlbin, "Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 218-241. MR**0448926 (56:7231)****[2]**J. R. Cannon & Y. Lin, "A priori error estimates for Galerkin methods for nonlinear parabolic integro-differential equations," Manuscript, 1987.**[3]**E. Greenwell Yanik & G. Fairweather, "Finite element methods for parabolic and hyperbolic partial integro-differential equations,"*Nonlinear Analysis Theory, Methods & Applications*, v. 12, 1988, pp. 785-809. MR**954953 (90e:65196)****[4]**M.-N. Le Roux & V. Thomée, "Numerical solution of semilinear integro-differential equations of parabolic type with nonsmooth data,"*SIAM J. Numer. Anal.*(To appear.) MR**1025089 (90m:65172)****[5]**I. H. Sloan & V. Thomée, "Time discretization of an integro-differential equation of parabolic type,"*SIAM J. Numer. Anal.*, v. 23, 1986, pp. 1052-1061. MR**859017 (87j:65113)****[6]**V. Thomée, "Negative norm estimates and superconvergence in Galerkin methods for parabolic problems,"*Math. Comp.*, v. 34, 1980, pp. 93-113. MR**551292 (81a:65092)****[7]**V. Thomée,*Galerkin Finite Element Methods for Parabolic Problems*, Lecture Notes in Math., vol. 1054, Springer-Verlag, 1984. MR**744045 (86k:65006)****[8]**M. F. Wheeler, "A priori error estimates for Galerkin approximations to parabolic partial differential equations,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 723-759. MR**0351124 (50:3613)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20,
65M60

Retrieve articles in all journals with MSC: 65R20, 65M60

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0969493-9

Keywords:
Integro-differential equation,
parabolic,
nonsmooth data,
regularity estimates,
finite elements,
Galerkin,
semidiscrete,
error estimates

Article copyright:
© Copyright 1989
American Mathematical Society