The numerical solution of elliptic partial differential equations

In this thesis numerical methods for solving elliptic partial differential equations are developed. These differential equations are discretized using finite differences and the resulting algebraic equations are then solved using iterative techniques. In particular, dynamic A.D.I. and multigrid methods are considered. These methods are tested on a model problem and then extended to solve more difficult problems. A coupled system of equations arising from solid state electronics is solved. Techniques are discussed to improve the accuracy of the resulting approximation near the two weak singularities the problem possesses. The dynamic A.D.I. and multigrid methods are extended to solve the biharmonic equation. A multigrid method which treats the biharmonic equation as a coupled system of two second order elliptic equations is also considered. A dynamic A.D.I. method for solving the 'double glazing' problem is presented. A non-uniform grid is used to resolve the boundary layers which develop for large values of the Rayleigh number.