Fast Uzawa algorithm for generalized saddle point problems
type: Item
instance of: scholarly article
title: Fast Uzawa algorithm for generalized saddle point problems
zbMATH Open document ID: 1032.65029
DOI: 10.1016/S0168-9274(03)00023-0
author: Zhi-Hao Cao
published in: Applied Numerical Mathematics
publication date: 2003-08-07T00:00:00Z
review text: Uzawa algorithms for solving generalized saddle point problems \[ \left( \begin{matrix} A & B^\top \\ B & -C \end{matrix} \right) \binom xy= \binom fg \] are considered. At first, convergence results for the inexact linear and nonlinear Uzawa algorithm for solving saddle point problems [see \textit{J. H. Bramble}, \textit{J. E. Pasciak}, and \textit{A. T. Vassilev}, SIAM J. Numer. Anal. 34, 1072-1092 (1997; Zbl 0873.65031)] are extended to the corresponding algorithms for generalized saddle point problems. Then, a new nonlinear Uzawa algorithm is presented in which a nonlinear approximation of the inverse of the Schur complement \((B A^{-1} B^\top + C)\) is used to accelerate the convergence. The convergence of this nonlinear Uzawa algorithm is analyzed. The Uzawa algorithms are applied to solve the incompressible steady state Stokes problem discretized by means of the mixed finite element method. The presented results of numerical experiments show that the new algorithm converges fast.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 76D07
Mathematics Subject Classification ID: 76M10
Mathematics Subject Classification ID: 65N12
zbMATH DE Number: 1961607
zbMATH Keywords: iterative methods
zbMATH Keywords: indefinite systems of linear equations
zbMATH Keywords: generalized saddle point problem
zbMATH Keywords: Uzawa algorithm
zbMATH Keywords: preconditioning
zbMATH Keywords: Stokes problem
zbMATH Keywords: mixed finite element method
zbMATH Keywords: convergence acceleration
zbMATH Keywords: numerical experiments
reviewed by: Michael Jung
MaRDI profile type: Publication
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full work available at URL: s0168-9274(03)00023-0
OpenAlex ID: W2029577565
description: scientific article; zbMATH DE number 1961607