Convergence Analysis for A Class of Iterative Methods for Solving Saddle Point Systems
For researchers solving saddle point problems, this offers relaxed convergence conditions, but the results are incremental extensions of existing theory.
The paper provides convergence analysis for nested iterative schemes (BWY and inexact Uzawa) for saddle point systems, establishing convergence under weaker conditions with specific contraction bounds (e.g., (√5-1)/2 for BWY, √2/2 for inexact Uzawa).
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.