A new finite difference representation for Poisson's equation on \(\mathbb R^3\) from a contour integral (Q618048)

The authors introduce a new finite difference approach for solving the \(3d\) Poisson's equation by representing the solution with contour integrals on local domains centered at each isolated grid node. The contour integrals are calculated exactly by assuming that the solution is a piecewise linear interpolation on each local triangular pyramid. The procedure gives a seven point representation of a finite difference type. The superconvergence of the scheme is established using a maximum principle and a priori estimate for the finite difference operator. A superconvergence property is attained when the solution \(u\) is in the function class \(C^{2,\alpha }\left( \overline{\Omega }\right) \cup C^3\left( \overline{\Omega }\right) ,\) \( 0<\alpha <1.\) Also, if \(u\in C^{3,1}\left( \overline{\Omega }\right) ,\) the standard \(O\left( h^2\right) \) convergence is obtained.