Limitations on numerical iterative solution of elliptic two dimensional partial differential equations

Abstract

Numerical iterative methods are applied for the solution of two dimensional Elliptic partial differential equations such as Laplace’s and Poisson’s equations. These kinds of differential equations have specific applications models of physics and engineering. The distinct approximation of the two equations is founded upon the theory of finite difference. In this work, the approximation of five point’s scheme of finite difference method is used for the equations of Laplace and Poisson to get linear system of equations. The solution of these Dirichlet boundary is discussed by finite difference method. An elliptic PDE transforms the PDE into a system of algebraic equations whose coefficient matrix has a tri-diagonal block format, using the finite difference method. Numerical iterative methods such as Jacobi method and GaussSeidel method are used to solve the resulting finite difference approximation with boundary conditions.