On the Numerical Fixed Point Iterative Methods of Solution for the Boundary Value Problems of Elliptic Partial Differential Equation Types

Abstract

In this research work, we have studied the finite difference method and used it to solve elliptic partial differential equation (PDE). The effect of the mesh size on typical elliptic PDE has been investigated. The effect of tolerance on the numerical methods used, speed of convergence, and number of iterations was also examined. Three different elliptic PDE’s; the Laplace’s equation, Poisons equation with the linear inhomogeneous term, and Poisons equations with non-linear inhomogeneous term were used in the study. Computer program was written and implemented in MATLAB to carry out lengthy calculations. It was found that the application of the finite difference methods to an elliptic PDE transforms the PDE to a system of algebraic equations whose coefficient matrix has a block tri-diagonal form. The analysis carried out shows that the accuracy of solutions increases as the mesh is decreased and that the solutions are affected by round off errors. The accuracy of solutions increases as the number of the iterations increases, also the more efficient iterative method to use is the SOR method due to its high degree of accuracy and speed of convergence.

Keywords: Partial differential equations, finite difference method, iterative methods, convergence