Schur complement based preconditioners are wellestablished and studied
for classical saddle point problems in $\mathbb{R}^N \times \mathbb{R}^M$.
In this thesis we extend these studies to multiple saddle point problems in
Hilbert spaces $X_1\times X_2 \times \cdots \times X_n$. For problems with
a block tridiagonal structure and a welldefined sequence of associated
Schur complements, sharp bounds for the condition number of the problems are
derived which do not depend on the involved operators. These bounds can be
expressed in terms of the roots of the difference of two Chebyshev
polynomials of the second kind.
This abstract analysis provides sufficient conditions for wellposedness.
This leads to new existence results and recovers known existence results
under less restrictive assumptions. It also provides a technique for
constructing parameter robust preconditioners. The method is applied to two
sets of problems:
 The classical threefield formulation of Biot’s consolidation model.
 Optimal control problems with a secondorder elliptic state equation.
Biot’s consolidation model has three model parameters. We derive a
preconditioner which is robust with respect to all of these parameters.
For the optimal control problems, we mainly consider problems with
distributed observation and limited control as well as distributed control
and limited observation. For both cases we provide existence results and
efficient preconditioners.
Due to a smoothness requirement, we use tensor product Bsplines in the
isogeometric framework for discretization. To efficiently realize the
derived preconditioners, we develop multigrid methods for biharmonic
problems. Two methods are proposed, one based on a GaussSeidel smoother
and one based on a mass smoother. Both methods are robust with respect to
mesh size and the later one is also robust with respect to spline degree.
By combining these smoothers, a hybrid smoother is created. This hybrid
smoother is numerically superior to the two other smoothers.
