Laplacian Filtered Loop-Star Decompositions and Quasi-Helmholtz Laplacian Filters: Definitions, Analysis, and Efficient Algorithms

TL;DR

This paper introduces filtered quasi-Helmholtz decompositions, including Loop-Star functions and Laplacian filters, which effectively manipulate operator spectra to improve preconditioning and solver efficiency in electromagnetic integral equations.

Contribution

It proposes new filtered quasi-Helmholtz decompositions capable of spectral manipulation, enabling the development of advanced preconditioners and fast solvers for electromagnetic problems.

Findings

Numerical results demonstrate improved preconditioning effectiveness.

Filtered decompositions effectively manipulate operator spectra.

Enhanced solver performance in electromagnetic integral equations.

Abstract

Quasi-Helmholtz decompositions are fundamental tools in integral equation modeling of electromagnetic problems because of their ability of rescaling solenoidal and non-solenoidal components of solutions, operator matrices, and radiated fields. These tools are however incapable, per se, of modifying the refinement-dependent spectral behavior of the different operators and often need to be combined with other preconditioning strategies. This paper introduces the new concept of filtered quasi-Helmholtz decompositions proposing them in two incarnations: the filtered Loop-Star functions and the quasi-Helmholtz Laplacian filters. Because they are capable of manipulating large parts of the operators' spectra, new families of preconditioners and fast solvers can be derived from these new tools. A first application to the case of the frequency and h-refinement preconditioning of the electric field integral equation is presented together with numerical results showing the practical effectiveness of the newly proposed decompositions.