Loop-star decomposition for any order basis functions with processing of weak and nearly singularities for the surface integral equation

A procedure to obtain div-conforming elements for the Surface Integral Equation, by rotating some popular curl-conforming bases used in the Finite Elements Method (FEM) is presented. The method maintains the Helmholtz decomposition for surface currents, for any order of bases functions and any curvature order of the domain. Thanks to the current decomposition, a frequency scaling avoids the frequency breakdown. For addressing the problem of the singular term in the integral equation, we use a coordinate transformation to cancel out the singular term. The singularity cancellation is carried out directly on a plane local domain. The procedure is compatible with any order of curvature of the elements and invariant with the order of the bases functions. The coordinate transformation adds a parameter suitable to treat the near-singularities.