Spectral analysis of the preconditioned system for the 3 x 3 block saddle point problem

In this work, we consider some preconditioning techniques for a class of 3 x 3 block saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and some other applications. We propose an exact block diagonal preconditioner for solving the symmetric saddle point problem and its nonsymmetric form. We show that the corresponding preconditioned systems have six different eigenvalues. For the needs of practical application, we also present a class of inexact block diagonal preconditioners for solving the saddle point problems. For the symmetric system, we estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrix, respectively. For the nonsymmetric system, we derive some explicit and sharp bounds on the real and complex eigenvalues. Numerical experiments are presented to demonstrate the effectiveness and robustness of all these new preconditioners.