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{\large Michele Benzi \\
{\bf Preconditioners for stabilized finite element discretizations of the steady incompressible Navier-Stokes equations}}

Mathematics and Science Center \\ Emory University \\ 400 Dowman Drive \\ Atlanta \\ GA 30322
\\
{\tt benzi@mathcs.emory.edu}\end{center}

In this talk we consider block preconditioners for stabilized
saddle point systems arising from incompressible flow problems.
Our focus is on block triangular preconditioners based on the
augmented Lagrangian formulation. These techniques have been
shown to be remarkably robust and effective when applied to
linear systems arising from inf-sup stable mixed finite element
discretizations of the Stokes and Oseen problems. Indeed, with an
appropriate choice of the augmentation parameter these
preconditioners result in rates of convergence that are
independent of the mesh size and largely insensitive to
the value of the viscosity; see [1,2].

In this talk we describe a (non-trivial) extension of the
augmented Lagrangian-based block preconditioners to the
case of stabilized finite elements, e.g., Q1-Q1 and Q1-P0
elements. We also describe an inexpensive technique for
estimating the optimal value of the augmentation parameter.
This technique is applicable to both stable and stabilized
finite element schemes.

This is joint work with Maxim Olshanskii (Moscow State
University) and Zhen Wang (Emory University).

References

[1] M. Benzi and M. A. Olshanskii, {\em An augmented Lagrangian-based
approach to the Oseen problem}, SIAM J. Sci. Comput., 28 (2006),
pp. 2095-2113.

[2] M. Benzi, M. A. Olshanskii and Z. Wang, {\em Modifed augmented
Lagrangian preconditioners for the incompressible Navier-Stokes
equations}, Int. J. Numerical Methods in Fluids, in press.


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