High-order quasi-Helmholtz Projectors: Definition, Analyses, Algorithms

TL;DR

This paper introduces high-order quasi-Helmholtz projectors that improve the stability and accuracy of high-order discretized EFIEs, avoiding complex topological computations and maintaining stable condition numbers at low frequencies.

Contribution

It extends quasi-Helmholtz projectors to high-order discretizations, eliminating the need for cycle detection and ensuring constant condition numbers at low frequencies.

Findings

Effective stabilization of high-order EFIEs demonstrated

Constant condition numbers achieved across frequency range

Numerical results confirm robustness in complex scenarios

Abstract

The accuracy of the electric field integral equation (EFIE) can be substantially improved using high-order discretizations. However, this equation suffers from ill-conditioning and deleterious numerical effects in the low-frequency regime, often jeopardizing its solution. This can be fixed using quasi-Helmholtz decompositions, in which the source and testing elements are separated into their solenoidal and non-solenoidal contributions, then rescaled in order to avoid both the low-frequency conditioning breakdown and the loss of numerical accuracy. However, standard quasi-Helmholtz decompositions require handling discretized differential operators that often worsen the mesh-refinement ill-conditioning and require the finding of the topological cycles of the geometry, which can be expensive when modeling complex scatterers, especially in high-order. This paper solves these drawbacks by presenting the first extension of the quasi-Helmholtz projectors to high-order discretizations and their application to the stabilization of the EFIE when discretized with high-order basis functions. Our strategy will not require the identification of the cycles and will provide constant condition numbers for decreasing frequencies. Theoretical considerations will be accompanied by numerical results showing the effectiveness of our method in complex scenarios.