Nonstandard Norms and Robust Estimates for Saddle Point Problems

Abstract

In this paper we discuss how to find norms for parameter-dependent saddle point problems which lead to robust (i.e., parameter-independent) estimates of the solution in terms of the data. In a first step a characterization of such norms is given for a general class of symmetric saddle point problems. Then, for special cases, explicit formulas for these norms are derived. Finally, we will apply these results to distributed optimal control problems for elliptic equations and for the Stokes equations. The norms which lead to robust estimates turn out to differ from the standard norms typically used for these problems. This will lead to block diagonal preconditioners for the corresponding discretized problems with mesh-independent and robust convergence rates if used in preconditioned Krylov subspace methods.