Analysis of relaxed nonlinear inexact Uzawa algorithm for symmetric saddle point problems

In this paper, we propose a relaxed nonlinear inexact Uzawa algorithm (RNIU) for solving the symmetric saddle point problems. It is an inner-outer iteration method with the inner iterations using variable accuracy for solving the approximate Schur complement system. The variable relaxation parameter is introduced to improve the convergence. We give the convergence analysis of this relaxed algorithm with variable inner accuracy, based on a simple energy norm. Sufficient conditions are given for the convergence of RNIU, which slightly improve the existing convergence results for the nonlinear inexact Uzawa algorithm with uniform inner accuracy in the literature. A practical approach for setting the variable relaxed parameters is proposed, and numerical experiments are given to illustrate the efficiency and sensitivity of RNIU.