Fast iterative solution of stabilised Stokes systems, part I: using simple diagonal preconditioners

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables. Iterative solution of such indefinite systems is feasible and is an attractive approach for large problems. The use of stabilisation methods for convenient (but unstable) mixed elements introduces stabilisation parameters. We show how these can be chosen to obtain rapid iterative convergence. We propose a conjugate gradient-like method (the method of preconditioned conjugate residuals) which 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. Here we discuss the simple case of diagonal preconditioning. Our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable. We demonstrate 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.