A variational functional for the finite element method

Finite element techniques have been applied to a wide variety of problems in electromagnetics, but are often handicapped by the appearance of spurious solutions. An analytical method is developed that focuses on the effective functional form as the fundamental cause underlying the difficulties with spurious solutions. By using analytical rather than numerical means, it is shown that the effective functional form allows for the existence of an improper gradient behavior in a general field expansion. In order to eliminate spurious solutions in the finite element method 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. Numerical results are obtained that demonstrate the effectiveness of the new functional to prevent spurious solutions.