# PhD Defence: Nonstandard Sobolev Spaces for Preconditioning Mixed Methods and Optimal Control Problems

**Nonstandard Sobolev Spaces for Preconditioning Mixed Methods and Optimal Control Problems**

The main focus of this thesis is on the construction of efficient solvers for two types of problems that fit into the class of PDE-constraint optimization:

Distributed optimal control problems with a tracking-type cost functional and linear state equations

Mixed methods for elliptic boundary value problems

A solution of an optimization problem can be computed via the first-order optimality conditions, also called the optimality system. For the type of problems considered here the optimality system is linear and has saddle point form. After discretization we end up with a large scale linear system (again in saddle point form) for which an efficient solver is required.

For the construction of an efficient solver we follow an approach which is called operator preconditioning.There efficient preconditioners are constructed based on the fact that the involved operator equation is well-posed in a Sobolev space $X$.

We present two techniques for finding this space $X$ for a problem in saddle point form:

Interpolation technique

Lagrangian multiplier technique

By means of this techniques we derive a new well-posed continuous mixed variational formulation for of the corresponding optimality system, for a mixed method of the first biharmonic boundary value problem and the distributed optimal control problem with time periodic Stokes equations. In addition, we demonstrate the two techniques on another mixed formulation for the biharmonic equation and the distributed optimal control problem with time periodic parabolic equation.

This techniques are demonstrated for four model problems:

The first and the second problem is a mixed method for the first biharmonic boundary value problem. The third and the fourth problem are problems of optimal control, distributed-time periodic Stokes control and distributed-time periodic parabolic control.

For the first biharmonic boundary value problem a well-posed continuous mixed variational formulation is derived, which is equivalent to a standard primal variational formulation on arbitrary polygonal domains.Based on a Helmholtz-like decomposition for an involved nonstandard Sobolev space it is shown that the biharmonic problem is equivalent to three second-order elliptic problems, which are to be solved consecutively. Two of them are Poisson problems, the remaining one is a planar linear elasticity problem with Poisson ratio 0. The Hellan-Herrmann-Johnson mixed method and a modified version are discussed within this framework. The unique feature of the proposed solution algorithm for the Hellan-Herrmann-Johnson method is that it is solely based on standard Lagrangian finite element spaces and standard multigrid methods for second-order elliptic problems. Therefore, it is of optimal complexity.

The continuous mixed variational formulation for the first biharmonic boundary value problem involves a non-standard Sobolev space for which a Helmholtz-like decomposition is presented. This allows a decomposition of the continuous problem in three second order elliptic problems, which are to be solved consecutively. The first and the last problem are Poisson problems with Dirichlet conditions and the second problem is a pure traction problem in planar linear elasticity with Poisson ratio $0$.

As discretization method the Hellan-Herrmann-Johnson method (non-conforming method in this setup) and a conforming modification are discussed. A discrete version of the Helmholtz decomposition allows as in the continuous case to solve the system consecutively, by solving the discretized version of the second order elliptic problems mentioned above. For this problems standard multilevel or multigrid methods are available, which results in optimal convergence behaviour could be shown.

For the distributed optimal control problem with time-periodic Stokes equations a well-posed continuous mixed formulation

of the corresponding optimality system is derived. Based on the involved parameter-dependent norms of the continuous problem, a practically efficient block-diagonal preconditioner is constructed, which is robust with respect to all model and mesh parameters. The theoretical results are illustrated by numerical experiments with the preconditioned minimal residual (PMINRES) method.

In addition we demonstrate the interpolation technique and Lagrangian multiplier technique for two further problems:

the Ciarlet-Raviart mixed method for the first biharmonic boundary problem

the distributed optimal control problem with time-periodic parabolic equations