Spectral Properties of the EFIE-MoM Matrix for Dense Meshes With Different Types of Bases

Abstract

The relationship between the properties of the basis functions employed to discretize the electric field integral equation (EFIE) and the eigenvalue spectrum of the resulting method of moments (MoM) matrix is studied. We concentrate on dense meshes, i.e., on complex geometries with large number of unknowns per wavelength, and/or disparate mesh cells sizes, and/or low frequency. These are typical occurrences in antennas, packaging and microwave circuits. We show that if the basis functions have separated Fourier spectra, the diagonal MoM matrix entries are close to the eigenvalues, and explain it in terms of the Fourier spectrum of the Green's function. For such a basis, diagonal preconditioning drastically reduces the condition number. This does not happen with sub-domain basis functions like the Rao-Wilton-Glisson (RWG), or with their linear combination into loop-tree or loop-star bases that are employed to solve low-frequency problems. Finally we analyze the properties of a multiresolution (MR) basis formed by linear combinations of RWG, but whose functions possess some degree of (Fourier) spectral resolution. We show that there is still correspondence between matrix diagonal, Green's function (Fourier) spectrum, and matrix eigenvalues. Diagonal preconditioning of the MR-MoM matrix causes the eigenvalues to cluster around 1, as it happens with the MoM matrix for the magnetic field integral equation. This strongly impacts on the EFIE matrix condition number and the speed of convergence of iterative solvers.