A stabilized formulation for the solution of the incompressible unsteady Stokes equations in the frequency domain

Abstract

A stabilized finite element method is introduced for the simulation of time-periodic creeping flows. This new technique, which is formulated in the frequency rather than time domain, strictly uses real arithmetics and permits the use of similar interpolation functions for pressure and velocity for ease of implementation. It involves the addition of the Laplacian of pressure to the continuity equation with a complex-valued stabilization parameter that is derived systematically from the momentum equation. The numerical experiments show the excellent accuracy and robustness of the proposed method in simulating flows in complicated and canonical geometries for a wide range of conditions. The present method is mass conservative up to the tolerance of the underlying linear solver. It significantly outperforms its temporal counterpart, namely the standard pressure stabilized Petrov-Galerkin method, by lowering the cost and solution turnover time of the simulated cases by several orders of magnitude.