An Inexact Uzawa Algorithm for Generalized Saddle-Point Problems and Its Convergence

TL;DR

This paper introduces an inexact Uzawa algorithm with adaptive relaxation parameters for efficiently solving generalized saddle-point problems across various applications, with proven convergence and demonstrated robustness.

Contribution

It presents a novel inexact Uzawa algorithm with two variable relaxation parameters that adapt during iterations, improving convergence without prior spectral estimates.

Findings

Algorithm converges for symmetric and nonsymmetric systems.

Numerical experiments confirm robustness and effectiveness.

No need for spectral estimates on preconditioned subsystems.

Abstract

We propose an inexact Uzawa algorithm with two variable relaxation parameters for solving the generalized saddle-point system. The saddle-point problems can be found in a wide class of applications, such as the augmented Lagrangian formulation of the constrained minimization, the mixed finite element method, the mortar domain decomposition method and the discretization of elliptic and parabolic interface problems. The two variable parameters can be updated at each iteration, requiring no a priori estimates on the spectrum of two preconditioned subsystems involved. The convergence and convergence rate of the algorithm are analysed. Both symmetric and nonsymmetric saddle-point systems are discussed, and numerical experiments are presented to demonstrate the robustness and effectiveness of the algorithm.