An Interior a Priori Estimate for Parabolic Difference Operators and an Application

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.