Multiresolution discrete finite difference masks for rapid solution approximation of the Poisson's equation

Abstract

The Poisson's equation is an essential entity of applied mathematics for modelling many phenomena of importance. They include the theory of gravitation, electromagnetism, fluid flows and geometric design. In this regard, finding efficient solution methods for the Poisson's equation is a significant problem that requires addressing. In this paper, we show how it is possible to generate approximate solutions of the Poisson's equation subject to various boundary conditions. We make use of the discrete finite difference operator, which, in many ways, is similar to the standard finite difference method for numerically solving partial differential equations. Our approach is based upon the Laplacian averaging operator which, as we show, can be elegantly applied over many folds in a computationally efficient manner to obtain a close approximation to the solution of the equation at hand. We compare our method by way of examples with the solutions arising from the analytic variants as well as the numerical variants of the Poisson's equation subject to a given set of boundary conditions. Thus, we show that our method, though simple to implement yet computationally very efficient, is powerful enough to generate approximate solutions of the Poisson's equation.