A Modified Nonlinear Inexact Uzawa Algorithm with a Variable Relaxation Parameter for the Stabilized Saddle Point Problem

This paper proposes a modified nonlinear inexact Uzawa (MNIU) algorithm for solving the stabilized saddle point problem, by introducing a variable overrelaxation parameter to speed up convergence. MNIU is an inner-outer iteration method with variable inner accuracy. We give a detailed error analysis for the convergence of MNIU, based upon a newly defined error norm which helps to handle variable inner accuracy for the Uzawa method. We also simply formulate the optimal overrelaxation parameter. Sufficient conditions are given for the convergence of MNIU. We show that MNIU converges in a relatively large range for the variable inner accuracy setting. For constant inner accuracy $\delta$, MNIU is convergent when $\delta<1$, too. Compared with the original nonlinear inexact Uzawa algorithm (NIU) that converges only for constant accuracy and $\delta<1/3$, this is a significant improvement. We also show a practical approach for estimating the optimal overrelaxation parameter for numerical computation. Numerical experiments of MNIU are given and compared with other methods. These results confirm the significant improvements of MNIU.