Finite element techniques have been applied to a wide variety of problems in electro magnetics, but have been handicapped by the appearance of spurious solutions. Both weighted residual methods and variational methods are the basic finite element techniques that are examined to establish a framework for the discussion of spurious solutions. A simple waveguide example is used to explore the fundamental problem with these spurious solutions. A method is developed that focuses on the functional form as the fundamental cause underlying the difficulties with spurious solutions. By using analytical rather than numerical means, it is shown that the solution form allows for the existence of an improper gradient behavior in a general field expansion. A new functional that satisfies Maxwell's equations and eliminates spurious solutions is introduced. This new functional is shown to be self-adjoint and positive definite, thus providing an error minimization. The analytical form as well as the finite element method is applied to demonstrate the robust nature of the functional. / Ph. D.