Toward a New Class of High-Frequency Fast Direct Solvers Via Operator Filtering: Preliminary Results on Convex and Non-Convex Geometries

We present a new class of fast direct solvers for integral equations in the high-frequency regime. By applying carefully designed regularization strategies, we derive integral equations that inherently support fast direct solvers. However, their approximation by means of the boundary element method introduces discretization errors that degrade the spectrum of the preconditioned matrices. Extraction, compression, and filtering techniques must be employed to compensate the undesired spectral deficiencies, resulting in spectrally equivalent linear systems that are well-suited for fast direct solvers.