18th GAMM-Seminar Leipzig on
Multigrid and related methods for optimization problems

Max-Planck-Institute for Mathematics in the Sciences
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  18th GAMM-Seminar
January, 24th-26th, 2002
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  Abstract Volker John, Fri, 16.00-16.25 Previous Contents Next  
  Non-nested multi-level solvers for mixed problems - numerical studies at the 3D Navier-Stokes equations
Volker John (Otto-von-Guericke-Universitšt Magdeburg)

The discretization of the Navier-Stokes equations by higher order finite elements may considerably improve the accuracy of computed flow parameters which are important in application, like the drag or lift coefficient. However, the solution of the arising discrete saddle point problems is in general much more complicated compared to saddle point problems coming from lower order discretizations.

We will present numerical studies of a multigrid approach for higher order finite element discretizations which uses on all coarser levels upwind discretizations of lowest order non-conforming finite elements. In this so-called multiple discretization multigrid method, the multigrid hierarchy possesses one level more than the geometric grid hierarchy since two discretizations are applied on the finest geometric grid. The higher order one forms the finest level of the multigrid hierarchy and a stabilized lowest order non-conforming finite element discretization forms the next coarser level.

The first study supports analytical results obtained for the multiple discretization multigrid method applied to the Stokes equations.

The efficiency of this multigrid approach is studied numerically at the 3D Navier-Stokes equations (flow through a channel around a cylinder). It is compared to the standard multigrid approach, which uses the same discretization on each level. Both multigrid methods are used as solver for the discrete saddle point problems and as preconditioner in a flexible GMRES solver.

This is joint work with Petr Knobloch, Gunar Matthies and Lutz Tobiska.

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