Complementary geometric formulations for electrostatics

The simultaneous use of a pair of complementary discrete formulations for electrostatic boundary value problems (BVPs) allows to accurately compute electromagnetic quantities, such as capacitance or electrostatic force with a minimum computational effort. In fact, the two formulations provide the upper and lower bounds for these quantities and their averages result quite close to the exact solution even for extremely coarse meshes. Despite the potential benefit to the many three-dimensional large-scale applications, taking advantage of this feature is not exploited in practice due to theoretical difficulties in the potential design. The aim of this paper is to fill this gap by rigorously introducing a pair of three-dimensional complementary geometric formulations to solve electrostatic BVPs on domains covered by conformal polyhedral meshes. In particular, an original formulation based on a vector potential is introduced by using cohomology theory with integer coefficients. It is shown how the so-called thick links are needed, which are representatives of the second cohomology group generators of the dielectric region. Two easy-to-implement graph-theoretic algorithms to automatically find such a basis with optimal computational complexity are described. Some benchmark problems are presented to show how the simultaneous use of both formulations yields to a sensible computational advantage. Therefore, solvers based on complementary formulations should be embedded in the next-generation of electromagnetic Computer-Aided Engineering (CAE) softwares.