A Generalized Theory of Higher Order Vector Finite Elements and Its Applications in Three-Dimensional Electromagnetic Scattering Problems

Abstract

ABSTRACT In this paper we present a generalized theory of higher order vector finite elements. This theory aims at generating tctrahedral Whitney elements with tangential continuity, having better orders of approximation compared to the well-known first order edge elements. The proposed methodology deals with some fundamental theoretical issues about Whitney elements, such as the proper definition of degrees of freedom and the correct modeling of the nullspace of the curl operator, corresponding to the class of irrotational fields. The whole set of properties is enforced to the shape functions, in order to produce their explicit expressions. The new second and third order vector finite elements are implemented in a formulation for 3-D electromagnetic scattering. A second order absorbing boundary condition for rectangular boundaries is used to truncate the infinite space. The numerical results show the improved accuracy of the approximation, obtained by the use of higher order Whitney elements, thus rendering them a promising tool for electromagnetic analysis.