Fast iterative solution of stabilised Stokes systems. I: Using simple diagonal preconditions. (English) Zbl 0776.76024

Wathen, Andrew; Silvester, David

SIAM J. Numer. Anal. 30, No. 3, 630-649 (1993).

The authors propose a conjugate gradient-like method (the method of preconditioned conjugate residuals) that is applicable to symmetric indefinite problems, describe the effects of stabilisation on the algebraic structure of the discrete Stokes operator, and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends. The simple case of diagonal preconditioning is discussed. The results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable. It is demonstrated that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete Laplacian are approachable for the Stokes problem.

MSC:

76D07	Stokes and related (Oseen, etc.) flows
76M10	Finite element methods applied to problems in fluid mechanics
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10	Iterative numerical methods for linear systems

Keywords:

method of preconditioned conjugate residuals; discrete Stokes operator; eigenvalue spectrum; convergence; diagonal preconditioning; stabilised mixed elements