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

• Published in: IEEE transactions on antennas and propagation Vol. 71; no. 12; p. 1
• Main Authors: Merlini, Adrien, Henry, Clement, Consoli, Davide, Rahmouni, Lyes, Dely, Alexandre, Andriulli, Francesco P.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.12.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Algorithms; Computational Physics; Decomposition; EFIE; Electric fields; Electric filters; Filtering theory; Information filters; Integral equations; Manganese; Matrix decomposition; Microwave filters; Operators (mathematics); Physics; Preconditioning; quasi-Helmholtz decompositions; quasi-Helmholtz projectors; Rescaling
• ISSN: 0018-926X; 1558-2221
• DOI: 10.1109/TAP.2023.3283043

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 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.