Zeroth-order divergence-complete discretizations of the EFIE at very low frequencies E. Ubeda, J. M. Tamayo, Juan M. Rius AntennaLab, Dept. of Signal Theory and Communications, Universitat Politècnica de Catalunya, UPC, Jordi Girona 1-3, 08034 Barcelona (Spain) E-mail: ubeda@tsc.upc.edu ; jose.maria.tamayo@tsc.upc.edu ; rius@tsc.upc.edu Abstract – We present the Self-Loop basis functions, an edge-oriented divergence-conforming set with zero charge density. These basis functions allow a rearrangement of the Linear-linear basis functions set to overcome the low-frequency breakdown of the Electric-field Integral Equation. I. INTRODUCTION Basis functions have been classified traditionally as divergence-conforming or curl-conforming. The divergence-conforming basis functions enforce normal continuity of the electric current across the edges arising in the discretization. They are well- suited for the discretization in Method of Moments (MoM) [1] of the Electric-field Integral Equation (EFIE). The Rao, Wilton and Glisson (RWG) [2] set carries out a constant expansion of the normal component of the current across the edges. The Linear-linear (LL) set [3-5] establishes a linear expansion of the normal component of the current along the edge, which doubles the number of unknowns. Even though the Linear-linear set carries out a higher-order expansion of the current, both RWG and LL sets produce a piecewise constant expansion of the charge density. A MoM-discretization of the EFIE with the traditional div-conforming basis functions [6] suffers from the so-called low-frequency EFIE breakdown. At very low frequencies, the contribution of the vector potential to the impedance matrix becomes negligible, according to the finite machine precision, compared with the scalar potential. This makes the discretization of the EFIE ill-conditioned and the solution inaccurate. For double machine precision and direct solution of the resulting matrix, the low- frequency breakdown appears when analyzing objects with sizes of the mesh cells below 10 -8  [7]. Divergence-conforming MoM-discretizations of the EFIE with stable condition number and accurate solution at very low frequency regime are obtained by the rearrangement of the original discretized space into the combination of a solenoidal subspace and a non-solenoidal, quasi-irrotational, subspace. The solenoidal subspace identifies the null space of the scalar potential of the EFIE and thus allows a stable discretization at very low frequencies. The Loop-Tree and Loop-Star basis functions represent a rearrangement of the RWG-space. The Loop basis functions, expanding the RWG-solenoidal space, are vertex-oriented for simply connected surfaces and have zero divergence. They capture all the closed paths derived from the RWG discretization. Either the Tree or the Star basis functions capture the remaining non-solenoidal space, whereby they have non-zero divergence. The Tree basis functions, normally defined over planar meshings, capture all the Tree-paths in the RWG discretization. The triangle-oriented Star basis functions are normally preferred for meshing 3D objects because they are easier to implement. There exist generalized Loop-Tree decomposition schemes for MoM-discretizations complete to higher divergence-orders [9]. However, since both RWG and LL basis functions are complete to 0 divergence- order, such schemes are not required. In this paper, we present a new scheme to obtain a stable discretization in Method of Moments of the EFIE with the LL discretization in the very low frequency regime. Our strategy is based upon combining a novel set of basis functions, the Self-Loop basis functions, with the well-known Loop-Star basis functions.