## 65K10 Optimization and variational techniques [See also 49Mxx, 93B40]

This thesis is concerned with the design and analysis of an interface fitted finite element strategy
for the simulation of free- and moving boundary problems with sharp interfaces. The presented
strategy falls into the class of interface tracking methods based on the Arbitrary Lagrangian-Eulerian (ALE) formulation. The main goal is to resolve the fundamental drawback of mesh
degeneration typical of moving mesh strategies while preserving all other benefits, in particular a
well-defined, accurate, and explicit representation of the moving geometry in space and time. This
representation allows for a straight-forward definition and implementation of problem tailored finite element spaces.
Parametric finite element spaces based on curved elements serve as the baseline of the presented
approach and are introduced and discussed in the context of a prototypical elliptic interface problem. A solution to the associated problem of generating admissible, interface fitted parametrizations is presented in terms of a variational mesh optimization approach. In this approach, a mesh
quality functional subject to an alignment constraint is minimized which leads to interface aligned,
optimal, and non-degenerate parametrizations.
For time-dependent problems, interface aligned parametrizations are constructed such that they
behave smoothly in time within each time slab, while a re-parametrization taking place in between
two successive time slabs is permitted. This strategy yields parametric finite element spaces that
are discontinuous in time. Consequently, a discontinuous Galerkin (dG(k)) approach in time is
used to construct fully discrete schemes. For the lowest order variant dG(0), an a priori error
analysis is conducted to show that the proposed strategy leads to convergence rates that are sub-
optimal by one order with respect to space in the worst case. A strategy to solve arising space-time
systems for higher-order dG(k) methods is proposed. This strategy is based on a transformation of
the space-time system into real block diagonal form, such that an efficient, preconditioned Schur
complement formulation for the arising 2 × 2 blocks can be used. It is shown that the preconditioned system possesses a condition number that is uniformly bounded by 2.
The last part of this thesis addresses the application of the strategy to free boundary problems. In
the context of a general flow problem in the presence of an interface force, governing equations
are introduced, a fully discrete space-time formulation is derived and an efficient first order in
time method is proposed. Two different application scenarios, a two-phase flow example with
surface tension, and a fluid-structure interaction problem are considered to validate and evaluate
the proposed approach, and to show its benefits and limitations.

We study the shallow water equations augmented with pollution transport in one space dimension on prismatic channels with nontrivial channel width. We use both, the characteristic and the weak formulation, to construct an explicit solution of the corresponding Riemann problem. This solution is an important ingredient for Godunov's finite volume scheme, which is used to discretize a real world sewer system into a network of finite volume elements. We especially focus on a finite volume model for the network vertices, which implies weak coupling conditions at each vertex and is based on the solution of a transjunctional Riemann problem. In order to develop a process model for sewer systems of practical relevance, we introduce generalized numerical flux functions, which allow the modeling of wave reflections at nonprismatic channel junctions and walls. The generalized numerical flux functions also enable the modeling of controllable special structures like pumps, weirs and valves. The resulting process model is especially suited for real world sewer systems and is mathematically represented by a system of ordinary differential equations. The controllable process model is subject to an optimal control problem for real-time application. We discretize the process model with a general linear one-step scheme in time and use adjoint calculus to provide explicit formulas for the gradient and Hessian of the time discrete cost functional. We verify our approach with a self-written C++ application, which is tailored to optimize finite volume networks in real-time, and provide numerical results for two sewer systems, which are both based on practical application. We apply a receding horizon strategy to both test cases and compare the real-time control results with the previously computed solutions of an offline control approach.