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

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- Math. Comp.
**25**(1971), 43-58 Request permission

## 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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## Additional Information

- © Copyright 1971 American Mathematical Society
- 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