This thesis addresses many issues associated with finite element techniques, concentrating on ideas that are not often emphasized in the literature. Pulling together the ideas of mesh generation, sparse storage solution techniques, and functional development, in a single volume, this work provides basic tools for implementation of finite element techniques for both static and dynamic problems in electromagnetics. An automatic mesh generation scheme is developed by forming a Delaunay triangulation. A storage technique will be presented and used In conjunction with a conjugate gradient method to solve linear systems of equations. Application to electromagnetic problems will be demonstrated in the static, quasi-static, and full-field regimes. Laplace's equation is solved for various transmission line geometries to obtain capacitance and characteristic impedance. A finite element model that is a full field expression of Maxwell's equations is developed through a novel variational formulation involving the method of Lagrange multipliers, with attention given to the physical basis of the obtained functional. This model is then applied to the problem of determining the propagation constant of a waveguide. / Master of Science