Scalar potentials and vector functions for the high-order modelling of singular wedge current

Abstract

The interest in incorporating edge conditions in method of moment (MoM) solutions dates back to the mid seventies of the last century, with more recent contributions in [1]-[4]. In particular, singular higher-order complete vector bases of the divergence-conforming kind have been presented in [3] and, in the following, we assume the reader to be comfortable with the definitions and the element representation given in [3, 5]. By using the same notation of [3], this paper presents a method that defines singular divergence-conforming vector basis functions in terms of potentials. Although this method is alluded in [3], it is not fully discussed there. The results reported below will permit one to appreciate the properties of the new singular divergence-conforming bases that we will present in a different future paper; these properties will turn out to be more satisfactory than those of the bases discussed here, or of the bases given in [3]. Another important purpose of this paper is to generalize to the (curved) edge-singularity elements that mesh a wedge structure in the vicinity of its edge the Meixner [6, 7] and the Motz-like approximations [8, 4]. This extension serves to clarify the numerical approximation inherent to the use of singular divergence-conforming bases. As said in [3], our singular bases contain as a subset the regular bases given in [5] as well as (at least) another Meixner subset that contains singular terms that model the singular and the nonsingular irrational algebraic terms of the Meixner or Motz-like series. For the sake of brevity, we discuss only the vector functions of the lowest-order Meixner subset. Extension to higher-order is possible by using the technique given in [3].