A discretisation method with the $H_{\rm div}$ inner product for electric field integral equations

TL;DR

This paper introduces a new discretisation method using the $H_{div}$ inner product for the electric field integral equation, effectively addressing low-frequency breakdown and improving computational efficiency with a novel preconditioning approach.

Contribution

It proposes a novel $H_{div}$ scalar product discretisation method and a new preconditioning technique to enhance accuracy and efficiency for EFIE at low frequencies.

Findings

Overcomes low-frequency breakdown in EFIE discretisation.

Achieves faster computation with the new preconditioning.

Validated through numerical experiments.

Abstract

A discretisation method with the $H_{\rm div}$ inner product for the electric field integral equation~(EFIE) is proposed. The EFIE with the conventional Galerkin discretisation shows bad accuracy for problems with a small frequency, a problem known as the low-frequency breakdown. The discretisation method proposed in this paper utilises the $H_{\rm div}$ scalar product with a scalar coefficient for the Galerkin discretisation and overcomes the low-frequency problem with an appropriately chosen coefficient. As regards the preconditioning, we find that a naive use of the widely-used Calderon preconditioning is not efficient for reducing the computational time with the new discretisation. We therefore propose a new preconditioning which can accelerate the computation successfully. The efficiency of the proposed discretisation and preconditioning is verified through some numerical examples.