COMPARISON OF JACOBI ITERATION AND GAUSS-SEIDEL ITERATION METHODS ON SOLVING TWO DIMENSIONAL LAPLACE EQUATION

Abstract

This study focuses on the numerical technique has been used to solve two dimensional Laplace equation of the form

+

= 0 with Dirichlet boundary conditions in a rectangular domain and focuses on certain numericaltechnique for solving partial differential equations in particular, the finite difference method. The aim of this research is intended to study the numerical solution of two- dimensional Laplace equation by comparing Jacobi iteration method with Gauss-Seidel iteration method. We can apply finite difference formula to evaluate numerically the solution of Laplace’s equation at the grid points. We comparing between finite difference method with true solution of U ∗ . We solve the value U at any interior nodes is the mean of the values at the four neighboring points and the two equations are standard and diagonal five- point formula for Laplace equation with order O(h 2 ). In problem-1 we seen that if the number of intervals in x-axis and y-axis in the given region increases, then the mesh size goes smaller and smaller and the truncation error also closer to the error tolerance. In problem-2 we seen that from the tables by using Jacobi iteration method to converges takes 34 iterations with error 10 -6 and Gauss-Seidel iteration method to converges takes 18 iterations with the same error 10 -6 . Then the comparison between the two iterations, the Gauss-Seidel iteration is faster convergence than Jacobi iteration, so the GaussSeidel iteration is the best one to converge. The values of iteration compared by using MATLAB software.