Convergence Analysis for A Class of Iterative Methods for Solving Saddle Point Systems

TL;DR

This paper provides a convergence analysis for a class of iterative methods solving saddle point systems, demonstrating weaker conditions for convergence and analyzing the effectiveness of preconditioned iterative methods.

Contribution

It establishes convergence conditions for the BWY iterative scheme and inexact Uzawa method under weaker assumptions than previously known.

Findings

BWY scheme converges under weaker contraction conditions

Inexact Uzawa method also converges with weaker bounds

Preconditioned GMRES with BWY preconditioner converges under realistic assumptions

Abstract

Convergence analysis of a nested iterative scheme proposed by Bank,Welfert and Yserentant (BWY) ([Numer. Math., 666: 645-666, 1990]) for solving saddle point system is presented. It is shown that this scheme converges under weaker conditions: the contraction rate for solving the $(1,1)$ block matrix is bound by $(\sqrt{5}-1)/2$. Similar convergence result is also obtained for a class of inexact Uzawa method with even weaker contraction bound $\sqrt{2}/2$. Preconditioned generalized minimal residual method using BWY method as a preconditioner is shown to converge with realistic assumptions.