Parallel alternating directions algorithm for 3D Stokes equation

We consider the 3D time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, written in terms of velocity and pressure. For this problem, a parallel algorithm, based on a recently proposed direction splitting approach, is applied. Here, the pressure equation is derived from a perturbed form of the continuity equation, where the incompressibility constraint is penalized in a negative norm induced by the direction splitting. The scheme used in the algorithm is composed of: (a) pressure prediction, (b) velocity update, (c) penalty step, and (d) pressure correction. In order to achieve good parallel performance, the solution of the Poison problem for the pressure correction is replaced by a solution to a sequence of one-dimensional second order elliptic boundary value problems (in each spatial direction). The efficiency and scalability of the proposed approach are tested on two distinct parallel computers and the experimental results are analyzed.