A new finite difference representation for Poisson's equation on R ³ from a contour integral

In this paper, we propose a new finite difference representation for solving a Dirichlet problem of Poisson's equation on R3. The key idea of the new approach is to represent the solution with a contour integral connecting the nodal values of each local domain centered at each isolated grid node, which is based on the boundary integral equation on the local domain, and calculate the contour integral using a piecewise linear interpolation of the solution. A superconvergence of the scheme is analyzed using a maximum principle and a priori estimate for the finite difference operator. The convergence behavior is comparable to that of standard finite difference methods on rectangle grids, and a superconvergence property is attained when the solution u is in the function class C2,α(Ω̄)∪C3(Ω̄), 0 < α < 1. Also, if u∈C3,1(Ω̄), the standard O(h²) convergence is obtained.