On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type

Author:
Nai Ying Zhang

Journal:
Math. Comp. **60** (1993), 133-166

MSC:
Primary 65M60; Secondary 35K05, 65M15

MathSciNet review:
1149295

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The subject of this work is the application of fully discrete Galerkin finite element methods to initial-boundary value problems for linear partial integro-differential 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 Crank-Nicolson scheme, and a third-order (1,2) Padé-type scheme, the specific quadrature rules analyzed are based on the rectangular, the trapezoidal, and Simpson's rule. For all the schemes studied, optimal-order 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. High-order regularity results with respect to both space and time are given for the solution of problems with smooth enough data.

**[1]**Paolo Acquistapace and Brunello Terreni,*Existence and sharp regularity results for linear parabolic nonautonomous integro-differential equations*, Israel J. Math.**53**(1986), no. 3, 257–303. MR**852481**, 10.1007/BF02786562**[2]**Garth A. Baker, James H. Bramble, and Vidar Thomée,*Single step Galerkin approximations for parabolic problems*, Math. Comp.**31**(1977), no. 140, 818–847. MR**0448947**, 10.1090/S0025-5718-1977-0448947-X**[3]**Philip Brenner, Michel Crouzeix, and Vidar Thomée,*Single-step methods for inhomogeneous linear differential equations in Banach space*, RAIRO Anal. Numér.**16**(1982), no. 1, 5–26 (English, with French summary). MR**648742****[4]**J. R. Cannon and Y. P. Lin,*A priori**error estimates for Galerkin methods for nonlinear parabolic integrodifferential equations*, manuscript, 1987.**[5]**Michel Crouzeix and Vidar Thomée,*On the discretization in time of semilinear parabolic equations with nonsmooth initial data*, Math. Comp.**49**(1987), no. 180, 359–377. MR**906176**, 10.1090/S0025-5718-1987-0906176-3**[6]**Jim Douglas Jr. and B. Frank Jones Jr.,*Numerical methods for integro-differential equations of parabolic and hyperbolic types*, Numer. Math.**4**(1962), 96–102. MR**0140192****[7]**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**[8]**Y. P. Lin, V. Thomée, and L. B. Wahlbin,*Ritz-Galerkin projections to finite element spaces and applications to integro-differential and related equations*, Technical Report 89-40, Mathematical Sciences Institute, Cornell University.**[9]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****[10]**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**[11]**Vidar Thomée,*Galerkin finite element methods for parabolic problems*, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR**744045****[12]**Vidar Thomée,*On the numerical solution of integro-differential equations of parabolic type*, Numerical mathematics, Singapore 1988, Internat. Schriftenreihe Numer. Math., vol. 86, Birkhäuser, Basel, 1988, pp. 477–493. MR**1022978**, 10.1007/978-3-0348-6303-2_39**[13]**-,*Numerical solution of integro-differential equations of parabolic type*, Institut Mathématique de Rennes, Université de Rennes I, 1990.**[14]**Vidar Thomée and Nai Ying Zhang,*Error estimates for semidiscrete finite element methods for parabolic integro-differential equations*, Math. Comp.**53**(1989), no. 187, 121–139. MR**969493**, 10.1090/S0025-5718-1989-0969493-9**[15]**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****[16]**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**[17]**N.-Y. Zhang,*On the discretization in time and space of parabolic integro-differential equations*, Dissertation, Chalmers University of Technology and the University of Göteborg, 1990.

Retrieve articles in *Mathematics of Computation*
with MSC:
65M60,
35K05,
65M15

Retrieve articles in all journals with MSC: 65M60, 35K05, 65M15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1993-1149295-4

Keywords:
Initial-boundary value problem,
parabolic,
partial,
integro-differential equation,
regularity,
finite element method,
fully discrete,
Galerkin approximation,
stability,
error estimate,
quadrature

Article copyright:
© Copyright 1993
American Mathematical Society