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Mathematics of Computation

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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
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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 $\beta \leq 2$. 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 $v,v \in {L_2}$, optimal $O({h^r})$ convergence for positive time is possible in general only if $r \leq 4 - \beta$. This depends on the fact that the exact solution is then only in ${H^{4 - \beta }}$.

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Keywords: Integro-differential equation, parabolic, nonsmooth data, regularity estimates, finite elements, Galerkin, semidiscrete, error estimates
Article copyright: © Copyright 1989 American Mathematical Society