Preconditioners and their analyses for edge element saddle-point systems arising from time-harmonic Maxwell’s equations

Abstract

We derive and propose a family of new preconditioners for the saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell’s equations in three dimensions. With the new preconditioners, we show that the preconditioned conjugate gradient method can apply for the saddle-point systems when wave numbers are smaller than a positive critical number, while the iterative methods like the preconditioned MINRES may apply when wave numbers are larger than the critical number. The spectral behaviors of the resulting preconditioned systems for some existing and new preconditioners are analyzed and compared, and several two-dimensional numerical experiments are presented to demonstrate and compare the efficiencies of these preconditioners.