Analytic preconditioners for decoupled potential integral equations and wideband analysis of scattering from PEC objects

TL;DR

This paper introduces new analytic preconditioners for potential integral equations that improve their conditioning across a wide frequency range, enabling efficient analysis of scattering from PEC objects with complex geometries.

Contribution

The paper develops novel Calderón-type identities and preconditioners for DPIE, VPIE, and SPIE that achieve wideband well-conditioning and rapid convergence for multi-scale PEC scattering problems.

Findings

Preconditioners significantly improve convergence at high frequencies.

The methods are effective for complex, multi-scale geometries.

Numerical examples validate the wideband robustness of the approach.

Abstract

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the ill-conditioning behavior of CFIE at high frequencies. This paper addresses this deficiency through new Calder\'{o}n-type identities derived from the Vector Potential Integral Equation (VPIE). We construct novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets.