The numerical solution of elliptic partial differential equations by novel block iterative methods

Partial differential equations occur in a variety of forms in many different branches of Mathematical Physics. These equations can be classified according to various criteria, which may include such formal aspects as the number of dependent or independent variables, the order of the derivatives involved and the degree of non-linearity of the equations. These equations can also be categorised according to the methods which are employed for solving the partial differential equations, or according to particular properties which their solutions may possess. In pure analysis, to solve partial differential equations we may use methods such as transformation or separation of variables. However, these methods can only be applied to very special classes of problems [WEINBERGER, 1965]. In practice, we employ numerical methods for the solution of such systems and although the types of methods used in numerical analysis of differential equations do not generally correspond with those used in mathematical analysis, both depend upon particular properties of the solution. This thesis is concerned with the numerical solution of certain types of partial differential equations and therefore, with practical problems which can be treated by certain numerical methods. In practice, the numerical methods for solving the differential problems depend upon the nature of other auxiliary conditions, such as boundary or initial conditions. Certain types of auxiliary conditions are suitable only for certain corresponding types of differential equations, and in general, physical problems suggest auxiliary conditions which 3 are suitable for the differential equations involved in the problem. If, the auxiliary conditions are specified in such a way that there exists one and only one solution (uniqueness) for the differential problem, and in addition, a small change in these given auxiliary conditions result in a small change in the solution (stability), then the problem is said to be well-posed. Since numerical methods are by nature approximate processes, however, these methods rarely produce exact solutions for a given problem. However, it can be shown [see STEPHENSON, 1970], that if the differential problem is well-posed then the solution of this problem is expected to be accurate..