On the Hierarchical Preconditioning of the PMCHWT Integral Equation on Simply and Multiply Connected Geometries

TL;DR

This paper develops a hierarchical basis preconditioning strategy for the PMCHWT integral equation applicable to both simply and multiply connected geometries, improving numerical stability and efficiency.

Contribution

It introduces a novel preconditioning approach for PMCHWT that overcomes limitations of EFIE-based methods, especially for multiply connected geometries.

Findings

Hierarchical preconditioning improves system conditioning.

EFIE-based diagonal preconditioners are ineffective for PMCHWT in complex geometries.

The proposed method is effective at low frequencies and with mesh refinement.

Abstract

We present a hierarchical basis preconditioning strategy for the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation considering both simply and multiply connected geometries.To this end, we first consider the direct application of hierarchical basis preconditioners, developed for the Electric Field Integral Equation (EFIE), to the PMCHWT. It is notably found that, whereas for the EFIE a diagonal preconditioner can be used for obtaining the hierarchical basis scaling factors, this strategy is catastrophic in the case of the PMCHWT since it leads to a severly ill-conditioned PMCHWT system in the case of multiply connected geometries. We then proceed to a theoretical analysis of the effect of hierarchical bases on the PMCHWT operator for which we obtain the correct scaling factors and a provably effective preconditioner for both low frequencies and mesh refinements. Numerical results will corroborate the theory and show the effectiveness of our approach.