Volume-surface integral equations with hybrid curl-conforming and divergence-conforming basis functions

The Volume-Surface Integral Equation (VSIE) has been successfully used in computational electromagnetics for complicated structures, especially for coated objects. Since the VSIE only requires a mesh on the object, it can be used to solve scattering problems of electrically large complicated objects with high accuracy when combined with the Fast Multipole Method (FMM). The VSIE method is characterized by the use of different integral equations on perfect electric conductor (PEC) surfaces and in penetrable objects. The equations are discretized with the extended Galerkin projection method. Usually, scattering problems involving PEC surfaces are solved using Surface Integral Equations (SIE), where the RWG basis functions are commonly used to represent the unknown surface current density. Scattering problems involving dielectric objects can be solved using either the SIE (piecewise homogeneous object only) or the Volume Integral Equation (VIE). When it comes to the selection of a basis for the VIE, both the divergence-conforming and curl-conforming basis functions can be used with no particular advantage provided by one over the other [1]. Divergence-conforming basis functions are usually used with electric flux density as unknowns, because the normal continuity of electric flux density can be enforced. Curl-conforming basis functions are used with electric field as unknowns since the tangential continuity of electric field across the mesh element is automatically enforced. Research on coated objects using VSIE with RWG basis on surface and divergence-conforming basis functions in dielectric volume can be found in [6].