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

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, and L. B. Wahlbin,*Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations*, SIAM J. Numer. Anal.**14**(1977), no. 2, 218–241. MR**0448926****[2]**J. R. Cannon & Y. Lin, "A priori error estimates for Galerkin methods for nonlinear parabolic integro-differential equations," Manuscript, 1987.**[3]**Elizabeth G. Yanik and Graeme Fairweather,*Finite element methods for parabolic and hyperbolic partial integro-differential equations*, Nonlinear Anal.**12**(1988), no. 8, 785–809. MR**954953**, 10.1016/0362-546X(88)90039-9**[4]**Marie-Noëlle Le Roux and Vidar Thomée,*Numerical solution of semilinear integrodifferential equations of parabolic type with nonsmooth data*, SIAM J. Numer. Anal.**26**(1989), no. 6, 1291–1309. MR**1025089**, 10.1137/0726075**[5]**I. H. Sloan and V. Thomée,*Time discretization of an integro-differential equation of parabolic type*, SIAM J. Numer. Anal.**23**(1986), no. 5, 1052–1061. MR**859017**, 10.1137/0723073**[6]**Vidar Thomée,*Negative norm estimates and superconvergence in Galerkin methods for parabolic problems*, Math. Comp.**34**(1980), no. 149, 93–113. MR**551292**, 10.1090/S0025-5718-1980-0551292-5**[7]**Vidar Thomée,*Galerkin finite element methods for parabolic problems*, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR**744045****[8]**Mary Fanett Wheeler,*A priori 𝐿₂ error estimates for Galerkin approximations to parabolic partial differential equations*, SIAM J. Numer. Anal.**10**(1973), 723–759. MR**0351124**

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

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

Additional Information

DOI:
http://dx.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