Numerical performance of smoothers in coupled multigrid methods for the parallel solution of the incompressible Navier-Stokes equations

In recent benchmark computations [Schäfer M, Turek S. The benchmark problem ‘Flow around a cylinder’. In Flow Simulation with High-Performance Computers II, Hirschel EH (ed.), vol. 52 of Notes on Numerical Fluid Mechanics. Vieweg: Wiesbaden, 1996; 547–566], coupled multigrid methods have been proven as efficient solvers for the incompressible Navier–Stokes equations. A numerical study of two classes of smoothers in the framework of coupled multigrid methods is presented. The class of Vanka-type smoothers is characterized by the solution of small local linear systems of equations in a Gauss–Seidel manner in each smoothing step, whereas the Brass–Sarazin-type smoothers solve a large global saddle point problem. The behaviour of these smoothers with respect to computing times and parallel overheads is studied for two-dimensional steady state and time-dependent Navier–Stokes equations.