A Survey on a Numerical Solution of Boundary Value Problems for Partial Differential Equations Using Multigrid Method

Abstract

The goal of this thesis is to present a survey on the use of multi grid method in the solution of the boundary-value partial differential equations (PDEs) numerically. The finite difference discretization technique has been introduced. Some classical numerical methods of solution for PDEs are presented showing their advantages & disadvantages. The concept of the multigrid method is introduced. The components of the multigrid algorithm and the advantages of this method over other numerical methods of solution are focused. The use of multigrid method in the solution of several PDEs in two-dimensional and three-dimensional spaces is presented.

In this thesis, a proposed multigrid algorithm has been developed for the solution of Poisson's equation on a unit circle subject to Dirichlet's boundary conditions using polar coordinates. A software program using FORTORAN95 is developed to solve this problem. This program can be simply modified to solve more general problems.