Uzawa methods for a class of block three-by-three saddle-point problems

In this work, we consider numerical methods for solving a class of block three-by-three saddle-point problems, which arise from finite element methods for solving time-dependent Maxwell equations and some other applications. The direct extension of the Uzawa method for solving this block three-by-three saddle-point problem requires the exact solution of a symmetric indefinite system of linear equations at each step. To avoid heavy computations at each step, we propose an inexact Uzawa method, which solves the symmetric indefinite linear system in some inexact way. Under suitable assumptions, we show that the inexact Uzawa method converges to the unique solution of the saddle-point problem within the approximation level. Two special algorithms are customized for the inexact Uzawa method combining the splitting iteration method and a preconditioning technique, respectively. Numerical experiments are presented, which demonstrated the usefulness of the inexact Uzawa method and the two customized algorithms.