High accuracy finite difference approximation to solutions of elliptic partial differential equations

Abstract

A flexible finite difference method is described that gives approximate solutions of linear elliptic partial differential equations, Lu = G, subject to general linear boundary conditions. The method gives high-order accuracy. The values of the unknown approximation function U are determined at mesh points by solving a system of finite difference equations L(h)U = I(h)G. L(h)U is a linear combination of values of U at points of a standard stencil (9-point for two-dimensional problems, 27-point for three-dimensional) and I(h)G is a linear combination of values of the given function G at mesh points as well as at other points. A local calculation is carried out to determine the coefficients of the operators L(h) and I(h) so that the approximation is exact on a specific linear space of functions. Having the coefficients of each difference equation, one solves the resulting system by standard techniques to obtain U at all interior mesh points. Special cases generalize the well-known 0(h(6)) approximation of smooth solutions of the Poisson equation to 0(h(6)) approximation for the variable coefficient equation -div(p grad[u]) + Fu = G. The method can be applied to other than elliptic problems.