# Schur complement preconditioners for multiple saddle point problems and applications

## MSc Jarle Sogn

**Nov. 13, 2018, 2:30 p.m. S2 054**

Schur complement based preconditioners are well-established 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 well-defined 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 well-posedness. 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 three-field formulation of Biot’s consolidation model.
- Optimal control problems with a second-order 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 B-splines 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 Gauss-Seidel 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.