High-Order Discrete Helmholtz Decompositions for the Electric Field Integral Equation

We develop a differential form based formalism to address the problem of low-frequency breakdown of the electric field integral equation (EFIE). Note, in this formalism we approximate the surface magnetic field, not the surface current as is conventionally the case. A discrete Helmholtz decomposition is achieved for both triangular and quadrilateral curvilinear meshes based on a star-cotree decomposition. These decompositions are based upon the construction of a canonical basis which ab-initio possess the required separation into irrotational and nonirrotational spaces. This makes the process of construction clear and generally applicable. The construction of appropriate bases is demonstrated for a range of interpolation orders. The effects of these constructions is demonstrated on a simple flat PEC plate problem