# 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 dierential equations. The mapping proper-

ties of the involved dierential 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.