Subject: Nonstandard Sobolev Spaces for Preconditioning Mixed Methods and Optimal Control Problems
Author: Dipl.-Ing. Wolfgang Krendl
Supervision: A.Univ.-Prof. Dipl.-Ing. Dr. Walter Zulehner
External referee: Prof. Dr. Volker Schulz (Universität Trier)

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

This techniques are demonstrated for four model problems:

  1. 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.
  2. 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.
  3. & 4. 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
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