  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.
