Abstract. As the sound speed is in nite for incompressible ows, computation of the pressure constitutes the sti est component in the time advancement of unsteady simulations. For complex geometries, e cient solution is dependent upon the availability of fast solvers for sparse linear systems. In this paper we develop a Schwarz preconditioner for the spectral element method using overlapping subdomains for the pressure. These local subdomain problems are derived from tensor products of one-dimensional nite element discretizations and admit use of fast diagonalization methods based upon matrix-matrix products. In addition, we use a coarse grid projection operator whose solution is computed via a fast parallel direct solver. The combination of overlapping Schwarz preconditioning and fast coarse grid solver provides as much as a fourfold reduction in simulation time over previously employed methods based upon de ation for parallel solution of multi-million grid point o w problems.