Calderón preconditioning approaches for PMCHWT formulations for Maxwell's equations

Abstract

Preconditioning methods based on Calderón's formulae for the Poggio-Miller-Chang-Harrington-Wu-Tsai formulations for Maxwell's equations in 3D are discussed. Five different types of formulations are proposed. The first three use different basis functions for surface electric and magnetic currents. The first type is a preconditioning just by appropriately ordering the coefficient matrix using the Gramian matrix as the preconditioner. Other two types utilise preconditioners constructed using matrices needed in the main fast multipole method algorithms. The fourth and fifth types are similar to the second and third types, but they use the same basis functions for both surface electric and magnetic currents. We make several numerical experiments with proposed preconditioners to confirm the efficiency of these proposed methods.AT À1 y ¼ b followed by x ¼ T À1 y:This approach is called method 1. This approach is based on the observation that (15) holds. 564 K. NIINO AND N. NISHIMURA

Quadrilateral Barycentric Basis Functions for Surface Integral Equations

“…However, if the quadrilateral is exactly a parallelogram, then the surface divergence is independent of the parameters . It is also found that (23) which indicates that the integration of the divergence of , viz., total surface charge on , is 4, independent of its shape or size. The factor 4 is rooted in the fact that the parameter space has an area of 4.…”

“…Interlocking mesh method [8] was an attempt in defining DBFs for (curvilinear) quadrilaterals. Rooftop BFs were investigated for rectangles [23]. However, these approaches have mesh restrictions and are not general.…”

Calderón preconditioning of PMCHWT boundary integral equations for scattering by multiple absorbing dielectric particles

“…Squaring the operator in this way regularizes the system by shifting the accumulation points of the spectrum away from the origin and taming the hypersingular component. For details we refer to [51,50,52], but the regularization relies on the following relations (contained in (30)-(31)):…”

“…In particular we note that in the Bempp implementation, the BC basic functions are ordered and rotated consistently with their RWG siblings. But we note that other discretizations are possible [50,52] and are just as easily implemented with the Bempp software package.…”

“…An alternative operator-based approach, studied e.g. in [46,47,48,49,50,51,52,53], is "Calderón preconditioning", which involves squaring the original boundary integral operator to regularize the problem at the continuous level, i.e. before discretization.…”