Local pressure-correction for the Navier-Stokes equations

This article presents a novel local pressure correction method for incompressible fluid flows and documents a numerical study of this method. Pressure correction methods decouple the velocity and pressure components of the time-dependent Navier-Stokes equations and lead to a sequence of elliptic partial differential equations for both components instead of a saddle point problem. In some situations, the equations for the velocity components are solved explicitly (with time step restrictions) and thus the elliptic pressure problem remains to be the most expensive step. Here, we employ a multiscale procedure for the solution of the Poisson problem related to pressure. The procedure replaces the global Poisson problem by local Poisson problems on subregions. We propose a new Robin-type boundary condition design for the local Poisson problems, which contains a coarse approximation of the global Poisson problem. Accordingly, no further communication between subregions is necessary and the method is perfectly adapted for parallel computations. Numerical experiments regarding a known analytical solution and flow around cylinder benchmarks show the effectivity of this new local pressure correction method.