Nonstandard Norms and Robust Estimates for Saddle Point Problems

ao.Univ.-Prof. Dipl.-Ing. Dr. Walter Zulehner

Jan. 18, 2011, 3 p.m. P 004

In important issue for constructing and analyzing effcient solvers for a saddle point problem is to understand the right mapping properties of the problem.
We will concentrate on saddle point problems which result from the discretization of a system of partial differential equations. The mapping properties of the involved differential operators usually suggest the right norms for the discrete problems leading to mesh-independent estimates. These norms are quite often (discrete versions of) standard norms in Lebesgue or Sobolev spaces.
If the saddle point problem contains critical parameters (like regulation parameters in optimal control problems) one would like to use norms leading to mesh-independent estimates which are also robust with respect to these critical parameters. Here standard norms usually do not the job.
In this talk we will discuss the construction of norms for saddle point problems which lead to robust estimates: Firstly, on a purely algebraic level, a characterization of such norms is given for a general class of symmetric saddle point problems. Then we will apply these results to a family of (time-independent) optimal control problems and show how to derive such norms for this family. Based on these norms robust solvers will be constructed by using multigrid techniques.