Laplacian Filters for Integral Equations: Further Developments and Fast Algorithms

TL;DR

This paper introduces an extension of Laplacian filtered quasi-Helmholtz decompositions to a basis-free setting, enabling efficient spectral analysis of electromagnetic integral operators and proposing a fast, effective preconditioning scheme.

Contribution

It extends Laplacian filtered decompositions to a basis-free framework and develops a fast algorithm for their evaluation, improving electromagnetic integral equation solutions.

Findings

The new method accurately analyzes operator spectra without Loop-Star decomposition.

The fast scheme achieves quasi-linear complexity in the number of unknowns.

Numerical results demonstrate the approach's effectiveness and theoretical predictions.

Abstract

This paper extends the concept of Laplacian filtered quasi-Helmholtz decompositions we have recently introduced, to the basis-free projector-based setting. This extension allows the discrete analyses of electromagnetic integral operators spectra without passing via an explicit Loop-Star decomposition as previously done. We also present a fast scheme for the evaluation of the filters in quasi linear complexity in the total number of unknowns. Together with the fact that only a logarithmic number of these filters are required for solving the h-refinement breakdown of electric field integral equation, this results in an effective preconditioner that rivals Calder\'on strategies in performance without relying on barycentric refinements. Numerical results confirm the theoretically predicted behavior and the effectiveness of the approach.