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Gunar Matthies, Thu, 11.3012.00

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A Multigrid Method for Incompressible Flow Problems Using Quasi Divergence Free Functions
Gunar Matthies (Univ. Bochum)
(joint work with F. Schieweck, Magdeburg)
We consider a finite element method of higher order on a quadrilateral or
triangular, hexahedral or tetrahedral mesh for solving the
Stokes or NavierStokes problem by means of
discontinuous elements for the pressure and suitable conforming
elements for the velocity such that the global and the local
infsup condition are satisfied.
Our goal is the construction of a multigrid solver for the corresponding
algebraic system of equations.
The idea of this solver is to switch inside of the multigrid method
to a new velocity basis which
leads to a reduced system with a much lower number of unknowns
as well as a very small number of couplings between
them.
We call this new basis quasi divergence free since most of the basis
functions are dicretely divergence free which implies that they do not
have any coupling to the pressure.
The quasi divergence free basis functions can be constructed locally during
the assembling process of the stiffness matrix.
We create a multigrid method for solving the reduced problem efficiently.
Since most of the velocity basis functions are completely decoupled from
the pressure we can construct a smoother with low computational
costs.
The efficiency of the new multigrid method compared with other known multigrid
solvers is demonstrated by numerical experiments for the Stokes problem.
It is shown how the ideas for the construction of a quasi divergence free
basis can be extended from the Stokes equations to the incompressible
NavierStokes equations in the stationary and nonstationary case.




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