On the Hierarchical Preconditioning of the Combined Field Integral Equation

TL;DR

This paper explores hierarchical preconditioning techniques for the CFIE, adapting methods from EFIE and proposing a new scheme for unstructured meshes that improves system conditioning.

Contribution

It introduces a novel preconditioning scheme for CFIE on unstructured meshes using Helmholtz projectors and hierarchical bases, enhancing numerical stability.

Findings

Numerical results confirm improved conditioning of the system.

The proposed scheme effectively preconditions CFIE without solenoidal bases.

The method extends hierarchical preconditioning to more general mesh types.

Abstract

This paper analyzes how hierarchical bases preconditioners constructed for the Electric Field Integral Equation (EFIE) can be effectively applied to the Combined Field Integral Equation (CFIE). For the case where no hierarchical solenoidal basis is available (e.g., on unstructured meshes), a new scheme is proposed: the CFIE is implicitly preconditioned on the solenoidal Helmholtz subspace by using a Helmholtz projector, while a hierarchical non-solenoidal basis is used for the non-solenoidal Helmholtz subspace. This results in a well-conditioned system. Numerical results corroborate the presented theory.