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In this work several time discretization techniques for capillary flows, i.e. either one-phase free
surface flow or two-phase flow, are compared. The focus is the development of methods that
are a) of higher order (i.e. at least second order), b) unconditionally stable and c) do not suffer
from much numerical dissipativity.
The mathematical model of the problem is initially described and a variational, dimensionless
formulation is given that can be used for both one- and two-phase capillary flows. The formula-
tion is in Arbitrary Lagrangian-Eulerian (ALE) coordinates so a problem-adapted moving mesh
can be used. Utilizing the Finite Element Method (FEM) the equations are discretized in space
and a differential algebraic matrix-vector formulation is derived.
Several fully implicit and linearly implicit time discretization techniques are introduced, their
properties with respect to stability, convergence, and dissipativity are discussed. These methods
are then applied to discretize the equations in time. Thus fully discrete equations are derived
that can be used for computer simulations.
Furthermore a space-time Galerkin approach is presented, its stability, convergence and
dissipativity properties are discussed as well. This method is then applied to the variational
formulation of the capillary flow problems and a fully discrete system is derived. For the
space-time approach an energy estimate is proved that establishes the unconditional stability
of the method by analytical means.
In an application chapter several one- and two-phase exemplary problems are introduced
and solved with the described time discretization techniques. Using Finite Element simulations
the convergence and dissipativity properties are numerically investigated.
A concluding comparison of the methods presents the advantages and drawbacks of all
methods that were compared.

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.

In this thesis, a-posteriori error estimates for time discretizations of the incompressible time dependent Stokes equations by pressure-correction methods are presented.
Pressure-correction methods are splitting schemes, which decouple the velocity and the
pressure. As a result the Stokes equations reduce to much easier schemes, which can
be solved by cost-efficient algorithm. In a first step, a-posteriori estimates for the instationary Stokes system, discretized by the two-step backward differential formula method
(BDF2), are presented. This allows to compare the a-posteriori estimators of the discretized Stokes system with the estimators of the pressure-correction scheme.
In the second part of the thesis rigorous proofs of global upper bounds for the incremen-
tal pressure correction scheme discretized by backward Euler scheme as well as for the
two-step backward differential formula method (BDF2) in rotational form are presented.
Moreover, rate optimality of the estimators are shown for velocity (in case of backward
Euler and BDF2 in rotational form) and pressure (in case of Euler). Computational
experiments confirm the theoretical results.

Transparent and conductive thin films find broad application in optoelectronic devices such as touchscreens and solar panels, and are intended to satisfy two opposite properties.
On the one hand, these films should appear transparent in the visible light, which means that a large amount of light can pass through such a film.
On the other hand, electric energy induced by an applied voltage should be transported with a low electric resistance.
Besides the transparent and conductive thin films, we want to consider particle monolayers as another class of photonic nanostructures.
The particle monolayers are utilized, for instance, to control the diffuse scattering behavior of photodetectors used in solar cells.
This optical property is quantified by the haze factor.
Experiments show that the design, which includes both the material composition and the overall shape of the photonic nanostructures, has a noteworthy influence on the performance with respect to the intended purpose.
The main objective of this thesis is to optimize the design of such photonic nanostructures with respect to transmission, conductivity and haze factor by changing the material, the shape and
the geometry using gradient-based algorithms.
Before the individual optimization problems are specified, analytical and numerical solution methods for the involved partial differential equations to determine the optical and electrical properties are discussed.
The electromagnetic scattering of a single spherical particle and assemblies of spherical particles is formulated in terms of fundamental solutions of Maxwell's equations, i. e. the vector spherical wave functions.
In this context, the order of convergence of dedicated errors is numerically studied with respect to various parameters.
In particular, the numerical evaluation of the haze factor for particle monolayers consisting of non-spherical particle is challenging and
a suitable numerical solution scheme has been developed.
For this purpose, the Finite Element Method and a spectral method based on vector spherical wave functions are combined to a two-stage hybrid simulation scheme in which computationally expensive tasks can be computed in a so-called offline stage.
Hence, sophisticated algorithms for the optimization of material, shape and geometry accomplish the gradient-based design optimization of photonic nanostructures.

In this thesis, a numerical method for the simulation of two–phase flow problems with a deformable interface and coupled species transport is investigated. The coupling of species concentration and fluid dynamics originates from a concentration–dependent interfacial tension coefficient – an effect known as concentration–induced Marangoni convection.
For this purpose, a mathematical model of a single drop was formulated using subspaces of the conventional function spaces, a problem–specific nondimensionalization, an accelerated frame of reference, and a divergence formulation of interfacial stresses.
The numerical method is based on a finite element method within an arbitrary Lagrangian– Eulerian framework. This class of methods results in an explicit interface representation, and forms the basis of the subspace projection method, a novel method for the implementation of various interface conditions in two–phase flow problems via problem–specific projections. Interface conditions that are investigated in this thesis comprise the flow around a rigid body, the simulation of spherically–shaped fluid particles, interface conditions of deformable drops with and without Marangoni convection, and the simulation of spherical stagnant caps – a limiting case observed in single drop flow under the presence of surfactants.
An essential feature of the subspace projection method is its ability to represent discontinuous functions, enabled by using a computational grid with doubled interfacial nodes. Spurious oscillations –often observed in two–phase flow problems where the pressure is approximated by globally continuous functions– are considerably reduced.
Concentration–induced Marangoni convection (resulting from a non–vanishing tangential gradient of interfacial stresses) is a challenging application in the numerical simulation of two– phase flows with a deforming interface. The implementation of interfacial stresses is based on a divergence formulation. This formulation accounts for normal and tangential stresses without the necessity of approximating surface derivatives of the interfacial tension.
A validation of the numerical method is performed for different axisymmetric single drop flow applications.
Different binary systems (single, buoyant, deformable drops with constant interfacial tension) are numerically investigated in terms of terminal rise velocities, drag coefficients, and deformation, comprising the range from low to high interfacial tension coefficients and drop diameters below the onset of shape oscillations. An excellent agreement of simulation results and experimental data is obtained. In case of terminal rise velocities, the deviation is below 4%.
Simulations of ternary systems (binary systems with species transport) are performed with and without Marangoni convection. In case of high Peclet numbers, finite element simulations often suffer from instabilities due to convection dominance. Stabilizing methods are investigated with special regard to two–phase flow applications.
Simulations of fluid dynamics and species transport in single drop flow applications like rigid particles, spherically shaped drops, spherical stagnant caps, and thermocapillary migration, show good to excellent agreement to correlations from literature in systems with moderate Peclet numbers. The deviation of the thermocapillary rise velocity is below 0.2% to the prediction by Young, Goldstein and Block (1959).
The ternary system toluene/acetone/water with concentration–induced Marangoni convection is numerically investigated and the results are compared to experimental data. In spite of the convection dominance of the corresponding species transport problem and the assumption of axisymmetry despite the inherent three–dimensional nature of the Marangoni effect, a reasonable qualitative agreement was achieved.

Atherosclerosis is a disease of the arteries which can cause a reduction or complete blockage of blood flow and may thus lead to heart attacks or strokes, two of the most common causes of death worldwide. This motivates mathematical modelling and simulation of the disease. The timescale of disease progression, driven by the growth of plaque in the artery wall over years, differs greatly from other processes which are assumed to be relevant, e.g. the shear stresses exerted by the blood flow on the artery wall, which oscillates with the heart beat every second. This prevents a direct numerical simulation of such models, since long timescales would have to be resolved with a very fine step size. The goal of this thesis is to construct approximations which are simpler to solve numerically, through the use of multiscale methods, and to prove quantitative convergence results. Motivated by a model by Yang et. al. (2015), we will investigate two simplified submodels of atherosclerosis for which a rigorous multiscale analysis is possible. The first model studies slow plaque growth coupled to fast oscillating shear stresses caused by the blood flow. Mathematically this is realized through a slow ordinary differential equation coupled to a fluid equation with rapidly oscillating boundary conditions and growth-dependent, non-cylindrical space-time domain. The second model investigates substances quickly advected through the artery but only slowly diffusing into the semi-permeable wall. It consists of a system of coupled advection-diffusion and diffusion-reaction equations. With a small parameter epsilon, which expresses the timescale separation, the behavior of the solutions to these models in the limit epsilon to 0 is investigated. Both models are singularly perturbed, meaning that their solutions converge to functions which solve a differential equation of different type. For the first model it will be shown that the solution converges with order O(epsilon) to the solution of a limit equation which averages the effect of a time-periodic fluid equation. The second model yields a limit consisting of a coupled advection and diffusion-reaction equation. The order of convergence depends on the solution regularity and the behavior of the advection field. For e.g. the stationary problem and Poiseuille flow it will be shown that the spatial L2- and H1-errors are of order O(epsilon^(1/2)), respectively O(epsilon^(1/6)), in the advection domain. Inside the wall the H1-error will be of order O(epsilon^(1/3)). The derivation of this result combines qualitative convergence theory for advection-diffusion equations in the vanishing diffusion limit with a specific trace estimate for the coupling through the permeable wall. Numerical calculations are carried out for both models. For the plaque growth the focus lies on the solution of the time-periodic Navier–Stokes equation which will be required for the limit system, an existing algorithm from the literature is improved here. Furthermore, the error of the time-discrete equation is analyzed, which quantifies and emphasizes how the errors made in the different solution steps must be balanced for efficiency. For the second model a discontinuous Galerkin discretization is proposed and the agreement between theoretical and numerical results shown.

Liquid crystals are materials which are characterized by mesomorphic states between ordinary liquids and solid crystals. Due to their wide range of applications in fields like photonics, optics, materials science, and biophysics, a lot of research on liquid crystals has been initiated in the last decades. In this thesis, we are concerned with two-phase flows of active liquid crystals which have the ability to convert energy from the local environment into mechanical work. This activity mechanism provides possibilities to model biological phenomena like the autonomous movement of cells.
In Chapter 2, we derive a new micro-macro model for two-phase flows of active liquid crystals which consists of Navier-Stokes equations for incompressible fluids which are coupled to a Smoluchowski equation and a phase-field equation. To take into account bending properties and energetic properties of biological structures like cell membranes the underlying energetic structure of the phase-field equation consists of the Willmore energy and a penalty energy for changes of the surface area of the interface. The Smoluchowski equation describes the evolution of a configurational density. Let us emphasize that we consider regimes with a high concentration of polymers; thus, our model takes into account the pairwise interaction of polymers and effects like dissipation due to friction.
Chapter 3 is devoted to the proof of the existence of global weak solutions. We first introduce a fully discrete finite element approximation of our model which is regularized from below and above. After establishing the existence of discrete solutions to the fully discrete scheme we pass to the limit as the spatial discretization parameter and the regularization parameter from below go to zero. This yields the existence of solutions to a discrete-in-time continuous-in-space approximation of our model. Thanks to further regularity results for the macroscopic polymer number density we are able to improve the regularity of the microscopic number density. After establishing time regularity results we pass to the limit in the discrete-in-time continuous-in-space approximation as the temporal discretization parameter goes to zero and the regularization parameter from above goes to infinity to prove that the limit functions are solutions to an unregularized weak formulation of our model in two and three space dimensions.
Chapter 4 is dedicated to numerical simulations of our fully discrete finite element scheme and its implementation in the in-house framework EconDrop of the group of Prof. Dr. Günther Grün. After comparing different approaches to solve the ill-conditioned phase-field equation of sixth order we provide simulations of self-driven active liquid crystalline droplets to present the full practicability of our scheme.