# On a Refinement-Free Calder\'on Multiplicative Preconditioner for the Electric Field Integral Equation

## TL;DR

This paper introduces a new Calderón preconditioner for the electric field integral equation that is refinement-free, stable for complex geometries, and enables efficient conjugate gradient solutions without additional discretizations.

## Contribution

A novel refinement-free Calderón preconditioner for EFIE that produces a Hermitian positive definite system, avoiding second discretizations and ensuring stability on multiply connected geometries.

## Key findings

- Preconditioner is effective for canonical problems.
- Demonstrates stability on multi-scale, realistic geometries.
- Enables efficient conjugate gradient solutions without additional discretizations.

## Abstract

We present a Calder\'on preconditioner for the electric field integral equation (EFIE), which does not require a barycentric refinement of the mesh and which yields a Hermitian, positive definite (HPD) system matrix allowing for the usage of the conjugate gradient (CG) solver. The resulting discrete equation system is immune to the low-frequency and the dense-discretization breakdown and, in contrast to existing Calder\'on preconditioners, no second discretization of the EFIE operator with Buffa-Christiansen (BC) functions is necessary. This preconditioner is obtained by leveraging on spectral equivalences between (scalar) integral operators, namely the single layer and the hypersingular operator known from electrostatics, on the one hand, and the Laplace-Beltrami operator on the other hand. Since our approach incorporates Helmholtz projectors, there is no search for global loops necessary and thus our method remains stable on multiply connected geometries. The numerical results demonstrate the effectiveness of this approach for both canonical and realistic (multi-scale) problems.