An overlapping Schwarz Method for spectral element solution of the incompressible Navier-Stokes equations

Efficient solution of the Navier-Stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems. For unsteady incompressible flows, the pressure operator is the leading contributor to stiffness, as the characteristic propagation speed is infinite. In the context of operator splitting formulations, it is the pressure solve which is the most computationally challenging, despite its elliptic origins. We examine several preconditioners for the consistent L²Poisson operator arising in the P_N- P_N-2spectral element formulation of the incompressible Navier-Stokes equations. We develop a finite element-based additive Schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior Gauss points. For large two-dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation.