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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.

Abstract. In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.

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In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented.