The convergence rate of the minimal residual method for the Stokes problem
The solution of the classical Stokes equation in fluid dynamics leads to the solution of a system \(Ax= b\) with \(A\) symmetric but indefinite. The iterative solution of this system by the minimal residual method converges like \(|r_k|\leq e_k|r_0|\), where \(r_k\) is the residual of step \(k\). The value \(e_k= \min_p \max_{x\in E} |p(x)|\), where \(p\) is a polynomial of degree at most \(k\), with constant term equal to 1. Thus means that it is the equioscillation value of the minimax polynomial for \(E\). This set \(E\) is an inclusion set for the eigenvalues of the iteration matrix, but which does not contain 0. This in this case \(E\) consists of at least one positive and one negative interval. There exist no explicit expressions for \(e_k\), but in this paper, the asymptotics for \(e= \lim_{k\to \infty} e^{1/k}_k\) are given in the form \(1- \alpha^{3/2}(bc/ad)^{1/2}+ O(\alpha^{5/2})\) when \(E= [- a, - b\alpha]\cup [c\alpha^2, d]\). Typically, \(\alpha\) is related to the discretization parameter \(k\) for the Stokes equation. There is no theoretical evidence that the estimate is optimal, but numerical experiments show that it is quite close.
scientific article; zbMATH DE number 802469