The purpose of this thesis is to present a unified treatment of numerical solutions to the elliptic partial differential equations, particularly the Laplace, Poisson, and Helmholtz equations, in homogeneous and inhomogeneous isotropic media in electrostatic, magnetostatic, and electromagnetic field problems, with particular attention given to iterative solutions. Another objective is to design a family of methods for accelerating the convergence of these solutions. The concept of the convergence acceleration is generalized to deterministic and stochastic vector sequences. Several hybrid methods, combining the finite difference and finite element approaches, are proposed. The methods have been tested either by test examples or by practical solutions to field problems.

Doctor of Philosophy (PhD)