Local Pressure Correction for the Stokes System

Abstract

Pressure correction methods constitute the most widely used solvers for the time-dependent Navier-Stokes equations. There are several different pressure correction methods, where each time step usually consists in a predictor step for a non-divergence-free velocity, followed by a Poisson problem for the pressure (or pressure update), and a final velocity correction to obtain a divergence-free vector field. In some situations, the equations for the velocities are solved explicitly, so that the most numerically expensive step is the elliptic pressure problem. We here propose to solve this Poisson problem by a domain decomposition method which does not need any communication between the sub-regions. Hence, this system is perfectly adapted for parallel computation. We show under certain assumptions that this new scheme has the same order of convergence as the original pressure correction scheme (with  global projection). Numerical examples for the Stokes system show the effectivity of this new pressure correction method. The convergence order $\mathcal O(k^{ 2})$ for resulting velocity fields can be observed in the norm $l^2(0,T ;L^2(\Omega))$.