Error analysis of the Penalty-Projection Method for the Time Dependent Stokes Equations

Full Title: Error analysis of the Penalty-Projection Method for the Time Dependent Stokes Equations

Abstract:

We address in this paper a fractional-step scheme for the simulation of incompressible flows falling in the class of penalty-projection methods. The velocity prediction is similar to a penalty method prediction step, or, equivalently, differs from the incremental projection method one by the introduction of a penalty term built to enforce the divergence-free constraint.

Then, a projection step based on a pressure Poisson equation is performed, to update the pressure and obtain an (approximately) divergence-free end-of-step velocity. An analysis in the energy norms for the model unsteady Stokes problem shows that this scheme enjoys the time convergence properties of both underlying methods: for low value of the penalty parameter r, the splitting error estimates of the so-called rotational projection scheme are recovered, i.e. convergence as δt2 and δt3/2 for the velocity and the pressure, respectively; for high values of the penalty parameter, we obtain the δt/r behaviour for the velocity error known for the penalty scheme, together with a 1/r behaviour for the pressure error.

Some numerical tests are presented, which substantiate this analysis.

Paper presented by: Jean-Marc Hérard