On the Numerical Treatment of Elliptic System of Partial Differential Equations

Abstract

The thesis introduces the numerical treatment of elliptic system of partial differential equations in the plane using the finite difference method. Two types of grids (square - triangular) are considered. The structure of the resultant algebraic system depends on the grid as well as the labeling of the grid points (the natural, the electronic, the RBG, and the spiral). Different forms of iterative methods (iterative methods without relaxation parameters, iterative methods with only one parameter, and iterative methods with two parameters) for solving linear algebraic systems are discussed. This thesis consists of five chapters, Arabic summary, and English summary.

Chapter One: Elliptic Partial Differential Equations in the Plane

A system of two partial differential equations in the plane is studied. Transformation of systems of partial differential equations into canonical forms which contain the smallest possible number of parameters (only two parameters instead of twelve) is obtained. Classification of systems of partial differential equations in the plane is given. The finite difference method is used to transform differential equations into algebraic systems. The algebraic structures of the resultant algebraic systems as well as the grid labeling are established. The square grid is discussed with three labeling techniques (the natural, the electronic, and the red-black order) in connection with the standard Poission’s problem.

Chapter Two: Variants of Successive Overrelaxation Techniques

Iterative methods for solving linear algebraic systems are established. Jacobi and Gauss Seidel methods are classified as iterative techniques without relaxation factors. SOR and KSOR methods are classified as iterative techniques with only one relaxation factor. Different forms of the modified successive overrelaxation methods (MSOR, MKSOR, MKSOR1, and MKSOR2) are introduced as iterative techniques with two relaxation factors. Iteration matrices and functional eigenvalue relations are given. Because of the fixed values of the spectral radii of Jacobi and Gauss Seidel iteration matrices, they are used to compare the convergence speeds with other methods and for the selection of relaxation parameters.