Nonstandard Norms and Robust Estimates for Saddle Point Problems

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

Jan. 18, 2011, 4 p.m. P 004

In important issue for constructing and analyzing ecient 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 dis-
cretization of a system of partial di erential equations. The mapping proper-
ties of the involved di erential 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.