The fundamental collocation method applied to the nonlinear poisson equation in two dimensions

Abstract

The fundamental collocation method is adapted to the nonlinear Poisson equation in two dimensions with mixed boundary conditions of the Dirichlet and Neumann type. The technique is an iterative collocation procedure which requires a representation of the boundary of a finite region by N points and of the interior by M points. The order of the problem as determined by the dimensions of the collocation matrices is N × N for each iteration. The method also employs an adjustable parameter S which can be used to check for stability. The accuracy and efficiency are shown to be quite good on three example problems, two of which are for heat-transfer and non-Newtonian laminar flow. Suggestions for improving the method are made.