Multiparametric vector finite elements: A systematic approach to the construction of three-dimensional, higher order, tangential vector shape functions

A systematic methodology for the construction of three-dimensional tetrahedral higher order vector finite elements is presented in this paper. Appropriate degrees of freedom describing tangential field components on interfaces and shape functions depending on a set of unknown coefficients are introduced. A decoupling procedure is applied to determine the shape functions. This leads to a whole family of parameter-dependent vector finite elements. However, concerning the irrotational fields to be modeled by the finite element, additional relations among the coefficients are introduced, which fully determines them. The whole procedure is described via the introduction of a new second order tetrahedral finite element. Third order tangential vector finite elements are also derived and the method could be easily generalized to higher orders. A numerical implementation of the second order element in a waveguide scattering problem is finally presented.