A Calderón multiplicative preconditioner for the electromagnetic Poincaré–Steklov operator of a heterogeneous domain with scattering applications

Abstract

In the context of hybrid formulations, the Poincaré-Steklov operator acting on traces of solutions to the vector Helmholtz equation in a heterogeneous interior domain with a smooth boundary is regularized by a well-known boundary integral operator related to the homogeneous exterior domain. For the first time, this property allows us to simultaneously construct a Calderón multiplicative preconditioner for the discretized operator and for a 3-D hybrid finite/boundary element method formulation, applicable to electromagnetic scattering problems. Numerical examples demonstrate the effectiveness of this novel preconditioning scheme, even for heterogeneous domains with non-smooth boundaries.

Magnetic and Combined Field Integral Equations Based on the Quasi-Helmholtz Projectors

“…(28) This equation is the magnetic field counterpart of the (localized) Calderón preconditioned electric operator in [37]. We propose to discretize (28) as…”

On a Low-Frequency and Contrast Stabilized Full-Wave Volume Integral Equation Solver for Lossy Media

“…Volume integral equations, instead, can model objects with a high degree of inhomogeneity. Unfortunately, as their surface counterparts, they suffer from the HC breakdown [19], [23]- [28] and fail to converge rapidly in applications with high-permittivity contrast scatterers. Another limitation of traditional VIE is that, even though they are immune from the low-frequency breakdown in purely dielectric objects [29], a frequency illscaling between the different parts of the VIE can occur when the object under study has a complex permittivity which depends on the frequency [30].…”

Electromagnetic modeling of high magnetic contrast media using Calderón preconditioning

“…In general, a faster convergence of the iterative solution is obtained when compared to alternative methods such as the finite element (FE) method, since only the surfaces of the scattering objects are discretized, leading to fewer unknowns. In case the problem includes heterogeneous regions, hybrid FE-BIE formulations may be adopted to take advantage of the efficiency of the BIE, and the ability of the FE method to model heterogeneous media [1].…”

“…The NtD operator maps the trace of the magnetic field on the boundary of a scattering object onto the tangential electric field on this boundary. The ill-conditioning of the discretization of Y will be resolved by making use of the regularizing property of the electric field integral operator T on the NtD operator [1]. The numerical solution of this novel formulation involves RWG and BC basis functions, which allows for integration in existing numerical solvers that make use of the conventional RWG basis functions.…”