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CHAPTER 1. INTRODUCTION 10
memory and CPU speed of today’s generation of computer hardware. A solution to this
problem are the algebraic multigrid (AMG) methods, where the initial mesh is used as
finest level, and the coarser levels are generated using (almost) only information of the
algebraic system.
A second reason for the popularity of AMG methods is their “black-box” character. In
an ideal situation the user does not need to construct any hierarchy, the method operates
on one single algebraic system and can therefore be used e.g. as a replacement for the direct
solver on the coarsest level of a geometric multigrid algorithm.
Since the pioneering work of Ruge and Stüben [RS86] and Brandt et al. [BMR84]
these methods have been applied to a wide class of linear systems arising (mostly) from
scalar partial differential equations. For an overview of the technique itself and various
applications we refer for example to Stüben [Stü01b].
For the application of AMG to saddle point problems one has the same two general
possibilities as in the geometric multigrid case. The first is the segregated approach, i.e. to
use a classical method (Uzawa, SIMPLE,. . . ) for an outer iteration and to apply AMG to
the resulting elliptic problems. This approach is described e.g. by Griebel et al. [GNR98]
or Stüben [Stü01a]. Another idea in this class is to use a Krylov space method such as
GMRES or BiCGstab with a special preconditioner which again decouples velocity and
pressure equations. This was done for example by Silvester et al. first for the Stokes case
[SW94] and later for the Navier-Stokes problem [SEKW01].
The focus of our work lies on the second possibility, on the coupled approach where
an AMG method for the whole saddle point system is developed (as mentioned above for
GMG methods). Work in this direction has been done for example by Webster [Web94] and
Raw [Raw95] for finite volume discretizations of the Navier-Stokes equations, by Bertling
[Ber02] for a finite element discretization of the Stokes equations, by Adams for contact
problems in solid mechanics [Ada03], and by Bungartz for constrained optimization (with
a small number of constraints) [Bun88].
This thesis is structured as follows. The second chapter contains the preliminaries which
are needed for a numerical solution of the Navier-Stokes equations. We start with the problem
statement, continue with the weak formulation and the finite element discretization,
sketch the analysis of the associated Stokes problem, mention some problems induced by
the convection, and finally discuss classical solution methods for the linear system.
In the third chapter we introduce algebraic multigrid methods. In this chapter we will
apply it only to scalar equations, but the underlying ideas will be important for the saddle
point case, too.
The central part of this work is chapter four, where we develop methods for the coupled
application of AMG methods to saddle point systems. We provide ideas for the construction
of multigrid hierarchies for different types of mixed finite elements, and we will deal with
stability problems which may occur on coarse levels. Unfortunately (but not surprisingly)
we were not able to construct a “black box method” capable of any saddle point problem,
with whatever choice of discretization on an arbitrary mesh. All our methods depend for
example on the concrete choice of the finite element.
Finally, chapter five is devoted to the presentation of numerical results. After a short