Uniform expansions for a class of finite difference schemes for elliptic boundary value problems

Abstract:

For a class of finite difference schemes for the Dirichlet problem on a bounded region $\Omega \subset {{\mathbf {R}}^n}$, the existence of uniform expansions of the approximate solution for meshlength $h \to 0$ is shown. The results also improve error bounds which Pereyra, Proskurowski, and Widlund obtained with respect to certain discrete ${L_2}$-norms.