Block triangular preconditioners for nonsymmetric saddle point problems: Field-of-values analysis
type: Item
instance of: scholarly article
title: Block triangular preconditioners for nonsymmetric saddle point problems: Field-of-values analysis
zbMATH Open document ID: 0922.65021
author: Axel Klawonn
author: Gerhard Starke
published in: Numerische Mathematik
publication date: 1999-05-31T00:00:00Z
review text: The problem considered is to construct a preconditioner \({\mathcal B}\) for the system \( {\mathcal A} x=f \) where \[ {\mathcal A}=\left(\begin{matrix} F&B^T\\ B&0 \end{matrix} \right) \quad \text{and}\quad {\mathcal B}=\left (\begin{matrix} F&B^T\\ 0&-S \end{matrix} \right) \] with \(F=\nu A+N\), \(A\) symmetric and \(N\) skew-symmetric, and \(S=BF^{-1}B^T\) is a Schur complement. Systems of this form arise for example in the solution of Navier-Stokes equations using finite elements. Since \({\mathcal B}^{-1}\) is needed, this requires the inverses of \(F\) and \(S\) which can only be computed approximately (inexact solves). This corresponds to using an approximate preconditioner \(\widehat{\mathcal B}\). Convergence results for GMRES both with the exact preconditioner \({\mathcal B}\) and with the approximate preconditioner \(\widehat{\mathcal B}\) are obtained. The upper bounds for the convergence rates are independent of \(h\) (the discretization parameter) and it depends only mildly on the viscosity \(\nu\).
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 1271905
zbMATH Keywords: GMRES
zbMATH Keywords: preconditioner
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: finite elements
zbMATH Keywords: saddle point problems
zbMATH Keywords: Schur complement
zbMATH Keywords: convergence
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
full work available at URL: s002110050405
OpenAlex ID: W2022665438
DOI: 10.1007/S002110050405
description: scientific article; zbMATH DE number 1271905