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

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 article introduces the new concept of filtered quasi-Helmholtz decompositions proposing them in two incarnations: 1) the filtered Loop-Star functions and 2) the quasi-Helmholtz 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 (EFIE) is presented together with numerical results showing the practical effectiveness of the newly proposed decompositions.