Constraint interface preconditioning for the incompressible Stokes equations

We introduce a novel substructuring approach for solving the incompressible Stokes equations for the case of enclosed flows. We employ a simple distribution of the global pressure constraint to subdomains which allows for a natural decomposition into Stokes subdomain problems with Dirichlet data which are well-posed and inf-sup stable. This approach yields a saddle-point problem on the interface Γ involving an operator which is continuous and coercive on H^1/2(Γ) and which is restricted to the interface trace space of functions satisfying the incompressibility constraint. We derive the form of the constraints explicitly, both for the continuous and for the discrete case. This allows us to design directly a class of interface preconditioners of constraint type, thus avoiding the need to formulate a coarse level problem. Our analysis indicates that the resulting solution method has performance independent of the mesh-size, while numerical results point to a mild dependence on the number of subdomains. We illustrate the technique on some standard test problems and for a range of domains, meshes and decompositions.