Discretization of Boundary Integral Equations by Differential Forms on Dual Grids

Abstract

In this paper, some integral equations of electromagnetics are reformulated in terms of differential forms. The integral kernels become double forms. These are forms in one space with coefficients that are forms in another space. The results correspond closely to the usual treatment, but are clearer and more intuitive. Since differential forms possess discrete counterparts, the discrete differential forms, such schemes lend themselves naturally to discretization. As an example, a boundary integral equation for the double curl operator is considered. The discretization scheme generalizes the well-known collocation technique by using de Rham maps. Depending on the integral operator to be discretized, the 1-form valued residual is forced to be zero either over the 1-chains of the primal or the dual grid.