Higher-order finite element discretizations in a benchmark problem for incompressible flows

Volker John, Gunar Matthies
2001 International Journal for Numerical Methods in Fluids  
We present a numerical study of several finite element discretizations applied to a benchmark problem for the two-dimensional steady state incompressible Navier-Stokes equations defined in Schäfer and Turek (The benchmark problem 'Flow around a cylinder'. In Flow Simulation with vol. 52, Hirschel EH (ed.). Vieweg: Wiesbaden, 1996; 547-566). The discretizations are compared with respect to the accuracy of the computed benchmark parameters. Higher-order isoparametric finite element
more » ... turned out to be by far the most accurate. The discrete systems obtained with higher-order discretizations are solved with a modified coupled multigrid method whose behaviour within the benchmark problem is also studied numerically. 886 capabilities of some discretizations and solvers. But a number of questions could not be answered definitely. One of these questions concerns the use of higher-order discretizations [1, p. 565, point 7]: The use of higher than second order discretizations in space appears promising with respect to the accuracy, but there remains the question of how to sol6e efficiently the resulting algebraic problems . . . . The results pro6ided for this benchmark are too sparse to allow a definite answer. In fact, even a second-order discretization was used only in one code. With this code, accurate results were obtained with a relative small number of degrees of freedom. Higher-order finite element discretizations for the incompressible Navier-Stokes equations are studied analytically, e.g. in [2] . Optimal order error estimates in norms of Sobolev spaces can be proven under certain conditions, especially under the assumption that the solution of the continuous problem is sufficiently smooth. However, this assumption will often not be fulfilled in practice. But on the other hand, norms of Sobolev spaces are in general not of practical interest. Quantities of interest in applications are e.g. mean values of the velocity, and drag or lift coefficients at obstacles. Systematic investigations of the accuracy of different discretizations with respect to such quantities seem to be rare. It is also not known how a missing global regularity of the solution affects the accuracy of higher-order discretizations for local quantities as drag or lift coefficients. This paper presents a numerical study for several finite element discretizations applied in the benchmark problem 'Flow around a cylinder'. The benchmark parameters are the drag and lift coefficient at the cylinder and the difference of the pressure between the front and the back of the cylinder. The numerical studies show that higher-order isoparametric finite element discretizations show the best accuracy with respect to the benchmark parameters despite the missing smoothness of the solution of the continuous problem. The discrete problems obtained with higher-order discretizations are solved by a modified coupled multigrid method with Vanka-type smoothers. The modification consists in using stabilized low-order finite element discretizations on the coarser multigrid levels. This solver is described in detail and its behaviour within the benchmark problem is studied for the higher-order finite element discretizations. THE BENCHMARK PROBLEM This section describes shortly the two-dimensional benchmark problem for the incompressible Navier-Stokes equations defined in [1] . We consider the stationary incompressible Navier-Stokes equations −wDu +(u·9)u +9p = 0 in V 9·u=0 in V u= g on (V with w =10 − 3 m 2 s − 1 and V is the channel shown in Figure 1 . The parabolic inflow and outflow profile
doi:10.1002/fld.195 fatcat:5ppyy564kbarbe3qwmzf7t2nka