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:
- 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.
- 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.
- & 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
- the distributed optimal control problem with time-periodic