Adaptive Uzawa algorithm for nonsymmetric generalized saddle point problem

TL;DR

This paper introduces a new adaptive Uzawa algorithm for nonsymmetric generalized saddle point problems that guarantees convergence without spectral assumptions, demonstrated through Navier-Stokes numerical experiments.

Contribution

A novel linear Uzawa type algorithm for nonsymmetric saddle point problems that converges independently of spectral estimates, improving robustness.

Findings

Algorithm converges without spectral assumptions.

Numerical tests on Navier-Stokes validate effectiveness.

Enhanced applicability to nonsymmetric problems.

Abstract

In this paper, we extend the inexact Uzawa algorithm in [Q. Hu, J. Zou, SIAM J. Matrix Anal., 23(2001), pp. 317-338] to the nonsymmetric generalized saddle point problem. The techniques used here are similar to those in [Bramble \emph{et al}, Math. Comput. 69(1999), pp. 667-689], where the convergence of Uzawa type algorithm for solving nonsymmetric case depends on the spectrum of the preconditioners involved. The main contributions of this paper focus on a new linear Uzawa type algorithm for nonsymmetric generalized saddle point problems and its convergence. This new algorithm can always converge without any prior estimate on the spectrum of two preconditioned subsystems involved, which may not be easy to achieve in applications. Numerical results of the algorithm on the Navier-Stokes problem are also presented.