Higher order nite element methods and multigrid solvers in a benchmark problem for the 3D Navier–Stokes equations

This paper presents a numerical study of the 3D ow around a cylinder which was de ned as a benchmark problem for the steady state Navier–Stokes equations within the DFG high-priority research program ow simulation with high-performance computers by Sch afer and Turek (Vol. 52, Vieweg: Braunschweig, 1996). The rst part of the study is a comparison of several nite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary tted higher order nite element methods are in general most accurate. Our numerical study improves the hitherto existing reference values for the benchmark parameters considerably. The second part of the study deals with e cient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The exible GMRES method with a multiple discretization multigrid method proves to be the best solver. Copyright ? 2002 John Wiley & Sons, Ltd.