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

Journal Article · Fri Jan 01 04:00:00 EST 1993 · Mathematics of Computation; (United States)

DOI:https://doi.org/10.1090/S0025-5718-1993-1149295-4· OSTI ID:6647492

Zhang, Naiying

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 Pade 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 Pade-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. 17 refs.

• OSTI ID: 6647492
• Journal Information: Mathematics of Computation; (United States), Journal Name: Mathematics of Computation; (United States) Vol. 60:201; ISSN 0025-5718; ISSN MCMPAF
• Country of Publication: United States
• Language: English

Related Subjects

661300* -- Other Aspects of Physical Science-- (1992-)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOUNDARY-VALUE PROBLEMS
CALCULATION METHODS
CONVERGENCE
DIFFERENTIAL EQUATIONS
EQUATIONS
EVALUATION
FINITE ELEMENT METHOD
GALERKIN-PETROV METHOD
ITERATIVE METHODS
NUMERICAL SOLUTION
PARABOLAS
PARTIAL DIFFERENTIAL EQUATIONS
QUADRATURES
SHAPE