An interior a priori estimate for parabolic difference operators and an application

Author:
Magnus Bondesson

Journal:
Math. Comp. **25** (1971), 43-58

MSC:
Primary 65.68

DOI:
https://doi.org/10.1090/S0025-5718-1971-0290598-9

MathSciNet review:
0290598

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Abstract | References | Similar Articles | Additional Information

Abstract: A general class of finite-difference approximations to a parabolic system of differential equations in a bounded domain $\Omega$ is considered. It is shown that if a solution ${U_h}$ of the discrete problem converges in a discrete ${L^2}$ norm to a solution *U* of the continuous problem as the mesh size *h* tends to zero, then the difference quotients of ${U_h}$ converge to the corresponding derivatives of *U*, the convergence being uniform on any compact subset of $\Omega$. In particular, ${U_h}$ converges uniformly on compact subsets to *U* as *h* tends to zero, provided there is convergence in the discrete ${L^2}$ norm. The main part of the paper is devoted to the establishment of an a priori estimate for the solutions of the discrete problem. This estimate is then used to derive the stated result.

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Keywords:
Parabolic difference operator,
a priori estimate,
boundary value problem,
convergence in maximum norm,
convergence of difference quotients

Article copyright:
© Copyright 1971
American Mathematical Society