Subject: Robust Multigrid Methods for Parameter Dependent Problems
Author: Dipl.-Ing. Joachim Schöberl
Supervisor: o.Univ.Prof. Dr. Ulrich Langer
External Referee: o.Univ.Prof. Dr. Dietrich Braess
 

This thesis is concerned with the construction and analysis of robust multigrid preconditioners for parameter dependent problems in primal variables. The developed framework is applied to the specific examples of nearly incompressibility, the Timoshenko beam, and the Reissner Mindlin plate. The suggested multigrid components are simple to implement.

solution of problem A

The work is based on two theories. On one side, the theory of mixed finite element methods is essential for the analysis of parameter dependent problems. The material is presented in a rather self-contained way. The point of departure is the primal form, where the stability conditions are formulated. The obtained discretization schemes are known as reduced integration methods. We point out the essence of a Fortin interpolation operator.

The other base contains the additive Schwarz theory and multigrid theory. These techniques provide a framework for the construction and analysis of preconditioners. We will collect the concepts of one-level and two-level subspace splitting, approximation property and smoothing property by means of elliptic problems without parameters. The formulation is such that it carries over to parameter dependent problems.

The central point of the thesis are the combination of both theories. We will start with one-level preconditioners for parameter dependent problems. We see that block smoothers capturing base functions of the kernel are robust with respect to the small parameter. The analysis uses function splitting with a partition of unity, and interpolation with the Fortin operator. We proceed with two level methods. A trivial prolongation by embedding is not uniformly bounded with respect to the parameter dependent energy norm. The idea for the construction of robust prolongation operators is that coarse grid kernel functions must be lifted to fine grid kernel functions. This can be obtained by adjusting proper degrees of fredoom locally. In the considered applications, the implementation consists of solving local sub-problems of the assembled stiffness matrix. The prolongation operator is an approximate right inverse of the Fortin operator.

solution of problem B

The components developed for the two-level method can be used in a multigrid algorithm. The analysis requires two norms, for which the approximation property and the smoothing property are verified. One is the parameter dependent energy norm, the other one combines three terms, namely improved convergence in a weaker norm of the primal variable, stability in primal energy, as well as stability in averae for the dual variable. The approximation property can be proven under abstract assumptions. The smoothing property has to be checked individually for the considered examples. Even under strongest realistic assumptions, we have to apply interpolation norms. We point out some relations to the smoother by Braess and Sarazin.

Finally, numerical experiments are presented. They agree with the analysis of the W-cycle and variable V-cycle. Additionally, they show optimal and robust convergence of V-cycle methods.

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