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Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations. (English) Zbl 1076.76544
Summary: This paper presents a numerical study of the 3D flow around a cylinder which was defined as a benchmark problem for the steady state Navier-Stokes equations within the DFG high-priority research program flow simulation with high-performance computers by M. Schäfer and S. Turek [Notes Numer. Fluid Mech. 52, 547–566 (1996; Zbl 0874.76070)]. The first part of the study is a comparison of several finite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary fitted higher order finite 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 efficient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The flexible GMRES method with a multiple discretization multigrid method proves to be the best solver.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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