Extrapolating solution of the laplace equation for planar field problem

This paper presents a method to find a solution of a two-dimensional field problem using finite-difference approach. The finite-difference technique has a slow convergence, but it is simple to program, and has a clear geometric interpretation. In this paper a thin conducting sheet was replaced by a mesh, with variable number of nodes. The potential of an arbitrary point of the sheet was approximated by a function having as arguments the order of the mesh, represented by the number of the nodes, and three unknown parameters. The problem was subsequently solved for several meshes with low number of nods, and the values of the parameters were found by a fitting procedure. The obtained approximating function predicted well the solutions for higher order meshes. There is a possibility that it can also predict the exact solution of the continuous problem seen as an infinite- order mesh for well-posed cases.