The nullspace method for the three-dimensional Stokes problem

In this paper, we present the preconditioned nullspace method for the iterative solution of the three-dimensional Stokes problem. In the nullspace method, the original saddle point system is reduced to a positive definite problem by representing the solution with respect to a basis of discretely divergence free vectors. The exact, explicit computation of such a basis typically has non-optimal (storage and computational) complexity. There exist some algorithms that exploit the sparsity of the matrix and work well for two dimensional problems but fail for three dimensions. Here, we will exploit an implicit representation of the nullspace basis which can be computed efficiently also in a three-dimensional setting, possibly only as an approximation. We will present some numerical results to illustrate the performance of the resulting solution method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)