CSCMO: Relative permittivity-based complex shifted operator preconditioning method for solving time-harmonic Maxwell equations

Abstract

In electromagnetic field modeling and simulation, a critical challenge lies in solving discretized time-harmonic Maxwell equations. The inherent complexity of these systems – including matrix indefiniteness and solution oscillations – poses significant difficulties for efficient numerical solutions. Preconditioned Krylov subspace methods have emerged as a standard approach for solving the large scale discretized time-harmonic Maxwell equations, where the construction of effective preconditioners remains pivotal. The shifted operator method represents a prominent preconditioning technique for Maxwell equations. Typically, a purely imaginary shift to the original differential operator is used, since this capitalizes on the observed phenomenon where solution oscillatory behavior diminishes with increasing modulus of the imaginary component of relative permittivity. In this paper, we propose a novel preconditioning technique by shifting both the real and imaginary parts of the relative permittivity. The motivation for proposing this kind of shifted operator preconditioning is based on theoretical analysis, which shows that decreasing the real part of the relative permittivity will improve the positive definiteness of the discretized matrix. The resulting preconditioned system admits efficient solution via multigrid methods. Some analysis shows that the spectral distribution of the preconditioned matrix is more clustered than the purely imaginary shift preconditioned matrix. Also, theoretical analysis for a special model shows that the condition number of the preconditioned system is less than its purely imaginary-shifted counterpart. Numerical results demonstrate that the proposed preconditioning is more effective than the other two state-of-the-art shift operator preconditioning methods.