Basic Iterative Methods for Solving Elliptic Partial Differential Equation

Abstract

ABSTRACT: In this thesis, we studied the numerical techniques for the solution of two dimensional Elliptic partial differential equations such as Laplace's and Poisson's equations. These types of differential equations have specific applications in physical and engineering models. The discrete approximation of both equations is based on finite difference method. In this research, five points finite difference approximation is used for Laplace's and Poisson's equations. To solve the resulting finite difference approximation basic iterative methods; Jacobi, Gauss-Seidel and Successive Over Relaxation (SOR) have been used. Several model problems are solved by three different iterative methods and concluding remarks are presented. Keywords: Elliptic partial differential equation, point's finite difference scheme, basic iterative methods.

………………………………………………………………………………………………………………………… ӦZ: Yapılan bu çalışma iki boyutlu Eliptik parçalı diferansiyel denklem problemlerinin sayısal analiz teknikleri kullanılarak yaklaşık olarak çözülmesi ile ilgilidir. Eliptik parçalı diferansiyel denklemler beş noktalı sonlu farklar yöntemi kullanılarak Poisson's ve Laplace denklemlerine uygulanmış ve bu denklemler temel iteratif çözüm prosedürü olan Jacobi, Gauss Seidel ve SOR yineleme yöntemleri kullanılarak iki farklı problem üzerinde nümerik olarak çözülmüştür. Ayrıca temel iteratif çözüm prosedürü teorik olarak incelenmiştir. Anahtar kelimeler: Eliptik parçalı diferansiyel denklem, beş noktalı sonlu farklar şeması, temel iteratif yöntemler.