Combined Projection Algorithms for Generalized Stokes Problems

Two projected methods have been proposed to deal with the two main diiculties of the numerical solution of incompressible Stokes problems. While one method enforces the null{ divergency condition through a projection 1], the other 4] projects the solution to satisfy boundary conditions. After summarizing these two techniques, the present report shows how to combine both into an accurate, eecient and simple generalized Stokes solver for large scale computations with a variety of complex and realistic boundary conditions. Preconditioning of the resulting projected system is handled by using specially designed matrices that share the same range and null space as the Boundary Condition (BC) projector Q and so commute with Q. The same idea can of course be applied to the single BC projection method introduced in 4], where the problem was left unsolved. One problem in implementing the double projection method concerns the rank-deeciencies introduced by the projector Q. In this paper we show how to eeciently compute the null{divergence (ND) projector P in presence of such rank deeciencies as well as the ones naturally found in CFD problems.