## 518 Numerische Analysis

Problems with free surfaces are ubiquitous in nature. The propagation of those surfaces is affected by various parameters, some of which are uncertain. Such effects can be interpreted as random noise and may lead to probabilistic terms in modeling [13, 15, 27, 34]. The scope of this thesis is to analyze the impact of stochastic terms in the Ito-sense on the propagation of solutions. In particular, we concentrate on porous-medium equations and parabolic p-Laplace equations.
Concerning stochastic porous-medium equations, we derive upper bounds on average waiting times and criteria for instantaneous propagation. Moreover, we present numerical experiments on average propagation.
Regarding stochastic degenerate-parabolic p-Laplace equations, we prove finite speed of propagation as well as sufficient conditions for the pathwise occurrence of waiting times. Afterwards, we derive a finite-element scheme for stochastic degenerate-parabolic p-Laplace equations that is nonnegativity preserving and also convergent. Finally, the quantitative propagation of solutions subjected to noise is investigated by means of numerical simulations.

The discontinuous Galerkin method for free surface and subsurface flows in geophysical applications
(2020)

Free surface flows and subsurface flows appear in a broad range of geophysical applications and in many environmental settings situations arise which even require the coupling of free surface and subsurface flows. Many of these application scenarios are characterized by large domain sizes and long simulation times. Hence, they need considerable amounts of computational work to achieve accurate solutions and the use of efficient algorithms and high performance computing resources to obtain results within a reasonable time frame is mandatory.
Discontinuous Galerkin methods are a class of numerical methods for solving differential equations that share characteristics with methods from the finite volume and finite element frameworks. They feature high approximation orders, offer a large degree of flexibility, and are well-suited for parallel computing.
This thesis consists of eight articles and an extended summary that describe the application of discontinuous Galerkin methods to mathematical models including free surface and subsurface flow scenarios with a strong focus on computational aspects. It covers discretization and implementation aspects, the parallelization of the method, and discrete stability analysis of the coupled model.

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.

This thesis deals with mathematical modeling, analysis, and numerical realization of microaggregates in soils. These microaggregates have the size of a few hundred micrometers and can be understood as the fundamental building units of soil. Thus, understanding their dynamically evolving, three-dimensional structure is crucial for modeling and interpreting many soil parameters such as diffusivities and flow paths that come into play in CO2-sequestration or oil recovery scenarios.
Among others, the following aspects of the formation of microaggregates should be incorporated into a mathematical model and investigated in more detail: the spatial heterogeneity of the temporally evolving structure of microaggregates and the different processes that take place on different scales — temporal and spatial — within the so-called micro-scale itself. This work aims at formulating a process-based pore-scale model, where all chemical species are measured in concentrations. That is, we have a continuous model for reactive transport mainly in terms of partial differential equations (PDEs) with algebraic constraints. This continuous model is defined on a discrete and discretely moving domain whose geometry changes according to the rules of a cellular automaton method (CAM). These rules describe the restructuring of the porous matrix, growth and decay of biomass, and the resulting topological changes of a wetting fluid and a gas phase. The cellular automaton rules additionally imply stochastic aspects that are important on the pore-scale.
Moreover, effects and knowledge deduced from the model are transfered to scales which are more relevant for applications. The quality of these averaged models is of general interest, since simulations for the field-scale that resolve the pore-scale are not applicable for economical reasons. Thus, this book compares parameterizations of diffusivities with mathematically rigorous results and gives suggestions to improve the formulas that can be found in the literature.
The discrete movement of the microaggregates’ geometry at the micro-scale poses mathematical problems. The following question arises: Can the averaged quantities deduced from the pore-scale really be used for models on other scales or are the impacts of the artificial temporal jumps too detrimental for the solutions on other scales to be accurate? In the following, this problem is also dealt with, and the reliability of the obtained parameters is underlined.
Last but not least, it is imperative to apply a proper numerical method to implement the model in silico. The local discontinuous Galerkin (LDG) method seems to be suitable for this task, since it is locally mass-conservative and is stable for discontinuous data — that might, for example, originate from the discrete movement of the geometry or from the sharp boundaries between the different phases. Additionally, this method has no problems with complicated transfer conditions. These aspects are demonstrated in a mathematically rigorous way, and the method is improved upon by reducing the linear system of equations resulting from the discretization. This is a real enhancement, since it does not diminish the order of convergence but decreases the computational costs.

This thesis is dedicated to the analysis, control and optimization for switched systems of abstract differential equations. A main focus lies on the special case of semilinear hyperbolic systems, including models describing the gas flow in pipe networks. First, a hierarchy of such models is introduced, together with the necessary graph-theoretical basics for the extension of these models to networks. We show how the models can be presented in a uniform formulation as semilinear hyperbolic initial boundary value problems. For systems of this kind a comprehensive solution theory is then developed, we prove the existence and uniqueness of solutions as well as their behavior, if the initial and boundary values show jumps. Furthermore, we consider the temporal switching between several such systems and prove a general result for the well-posedness of feedback-controlled switching processes. The example of gas networks with active elements such as valves, check valves and compressors demonstrates the achieved results. In a more abstract framework, we formulate an optimization problem for switched systems of evolution equations with strongly continuous semigroups. We specify optimization criteria and formulate an adjoint-based calculus that allows an efficient evaluation of gradients. The results are embedded in an alternating-direction-method, to which we present suitable convergence concepts. In addition to other applications in the field of ordinary, delayed and partial differential equations, we again cover the example of the optimization of gas networks. To this end, the numerical implementation is discussed, especially methods and schemes used for the simulation and optimization of gas networks. We present two application examples showing the suitability of our methods for the optimal switching of active elements in gas networks as well as optimized model selection balancing accuracy with computational effort.

In this thesis, we consider model reduction for parameter dependent parabolic PDEs defined on networks with variable composition. For this type of problem, the Reduced Basis Element Method (RBEM), developed by Maday and Rønquist, is a reasonable choice as a solution on the entire domain is not required. The reduction method is based on the idea of constructing a reduced basis for every individual component and coupling the reduced elements using a mortar-like method. However, this decomposition procedure can lead to difficulties, especially for networks consisting of numerous edges. Due to the variable composition of the networks, the solution on the interfaces is extremely difficult to predict. This can lead to unsuitable basis functions and poor approximations of the global solutions.
On the basis of networks consisting of one-dimensional domains, we present an extension of the RBEM which remedies this problem and provides a good basis representation for each individual edge. Essentially this extension makes use of a splinebased boundary parametrization in the local basis construction. To substantiate the approximation properties of the basis representation onto the global solution, we develop an error estimate for local basis construction with Proper Orthogonal Decomposition (POD) or POD-Greedy. Additionally, we provide existence, uniqueness and regularity results for parabolic PDEs on networks with one-dimensional domains, which are essential for the error analysis.
Finally, we illustrate our method with three examples. The first corresponds to the theory presented and shows two different networks of one-dimensional heat equations with varying thermal conductivity. The second and third problem demonstrates the extensibility of the method to component based domains in two dimensions or nonlinear PDEs. These were parts of the research project Life-cycle oriented optimization for a resource and energy efficient infrastructure, funded by the German Federal Ministry of Education and Research.

Stabilization Techniques for the Finite Element Method Applied to Advection Dominated Problems
(2018)

This thesis in the mathematical field of numerics of partial differential equations deals with different stabilization techniques of finite element discretizations of advection-dominated problems. The underlying advection–diffusion equation is used, e.g., in hydrogeology, where it describes the transport of groundwater-dissolved matter through the porous soil matrix at an averaged scale. The transport is caused by the actual groundwater flow ("advection") as well as by Brownian particle motion and grain-structure-related dispersion ("diffusion"). Discretizing the advection–diffusion equation in space using the classical finite element method (FEM), unphysical oscillations occur if advection dominates diffusion. The quotient of these two quantities defines the Peclet number. For high Peclet numbers, the numerical solution also attains negative concentration values.
Three different stabilization techniques are considered in this thesis: The first one uses a finite volume discretization for the advective part. For equations of hyperbolic character, finite volume methods are known to have better properties than the FEM. The second method is the streamline diffusion method (streamline upwind Petrov–Galerkin, SUPG) for polynomial degrees one and two. Finally, a not yet widely-established method is considered: Algebraic flux correction (AFC). AFC limits the mass flux between degrees of freedom such that a given prescribed principle is preserved. For example, one can guarantee the discrete non-negativity of concentrations using the AFC method. This is desirable as negative concentrations are physically not meaningful. AFC is formulated on the algebraic level, not on the variational one, and introduces additional non-linearities. Those can be handled by a fixed-point iteration or an appropriate linearization. In any case, AFC turns out to have the highest computational costs compared to the other two stabilization methods. While one can apply SUGP for arbitrary polynomial degrees, the finite volume stabilization and AFC in the standard "linear mass lumping" version can only be applied using first-order polynomials.

Ziel dieser Arbeit ist die effiziente L\"osung hochdimensionaler elliptischer partieller Differentialgleichungen auf d\"unnen Gittern.
Bei Diskretisierung wird der Galerkin-Ansatz und die Finite-Elemente-Methode verwendet.
Eine effiziente Implementierung der Matrix-Vektor-Multiplikation skaliert linear mit der Anzahl der Gitterpunkte.
Dazu wird ein Algorithmus vorgestellt, der durch eine Kombination von Restriktionen und Prolongationen die hierarchischen \"Ubersch\"usse aller Gitter einsammelt und verteilt.
F\"ur d\"unne Gitter muss allerdings auf den Austausch zwischen einigen Gittern verzichtet werden.
Dies hat aufgrund der Semi-Orthogonalit\"at keinen Einfluss auf die Konvergenz, da Prewavelets f\"ur \"uberlappende Gebiete $L^2$-orthogonal sind.
Bei variablen Koeffizienten zeigt ein Konvergenzbeweis, dass diese Werte sehr klein sind und keinen Einfluss auf die Konvergenz haben.
Die Konvergenzordnung der D\"unngitterdiskretisierung reduziert sich im Vergleich zu vollen Gittern nur leicht, w\"ahrend die Anzahl der Unbekannten dramatisch abnimmt.
Numerische Ergebnisse mit variablen Koeffizienten zeigen optimale Konvergenz f\"ur dreidimensionale und sechsdimensionale Probleme.
Transformationen f\"ur krummlinig umrandete Gebiete besitzen variable Koeffizienten und beruhen nicht auf einem Tensorprodukt.
Die Berechnung der Steifigkeitsmatrix verwendet eine hochdimensionale numerische Integration.
Multiplikationen werden durch Rekursion auf eindimensionale Operatoren zur\"uckgef\"uhrt.
Dazu ist im Rahmen dieser Arbeit eine umfangreiche Softwarebibliothek entstanden.
Durch die Verwendung von Templates kann die Implementierung auf beliebige hochdimensionale Probleme angewendet werden.
Parallelisierung sowohl f\"ur geteilten Speicher als auch verteilte Systeme gew\"ahrleistet eine hohe Genauigkeit f\"ur gro\ss{}e Dimensionen.
Ein Ansatz mit semi-adaptiver Verfeinerung erm\"oglicht partielle Verfeinerung bei gleichzeitigem Erhalt der Tensorprodukt-Struktur.
Die Umsetzung adaptiver d\"unner Gitter wird als Ausblick behandelt.

Nonlocal balance laws are nonlinear partial integro-differential equations that play a major role in the modeling of real world phenomena.
From the description of ripening processes in nanoparticle synthesis up to macroscopic modeling of vehicular traffic flow these equations are of crucial importance.
In the presented work a general formulation of one-dimensional nonlocal balance laws was analyzed with respect to existence, uniqueness and regularity of weak solutions.
The Kruzkov entropy condition is in the context of balance laws widely used to obtain uniqueness of weak solutions. The presented work shows that the entropy condition is obsolete in the discussed class of nonlocal balance laws.
An analytical representation of the weak solution is derived based on the method of characteristics and a fix-point mapping in the function space. The solvability of the fix point mapping for small times is proven by Banach's fixed-point theorem. A time-horizon where the weak solution exists is determined by clustering of these small times steps. The gained result is shown to be sharp in special cases.
Commonly weak solutions to nonlocal balance laws were approximated numerically using problem specific finite volume schemes. To get rid of the inherent numerical dissipation
an alternative numerical scheme is deduced based on the analytical representation of the weak solution. Therefore, a semi-discretization based on the method of characteristics and a piecewise constant representation of the solution is introduced and analyzed with respect to convergence.
In dependence on the global as well as piecewise regularity of the data a priori error estimates are derived. Therefore the whole spectrum from inital data of bounded variation up to Lipschitz-continuous initial data were analyzed.
The presented numerical scheme and parts of the analytical weak solution are applied to examples from the modeling of traffic flow and the nanoparticle synthesis. Thereby the high accuracy of the presented scheme is discussed in comparison to published simulation results. Through the introduced numerical method, discontinuities of the initial data are tracked over time and will not be smoothed, thus the character of the solution is represented more accurately. In the context of nanoparticle synthesis, optimal process conditions which lead to particle size distributions with small dispersity were determined exemplarily.

This thesis presents the three-dimensional modeling, discretization, implementation, and simulation of additive manufacturing processes on the example of electron beam melting (EBM). The fluid dynamics of the liquified melting pool are modeled by the incompressible Navier-Stokes equations and the incorporation of energy by the heat equation. The applied numerical scheme is a thermal multi-distribution lattice Boltzmann method (LBM) allowing an efficient parallel implementation. The liquid phase of the melting pool and the gas phase of the atmosphere are separated by the free surface lattice Boltzmann method (FSLBM) that does not compute the dynamics of the gas phase explicitly but sets a boundary condition at the interface. Furthermore, the electron beam gun and the metal powder particles are explicitly modeled. A realistic particle size distribution is achieved by using an inverse Gaussian distribution. For the absorption of energy two different algorithms are derived depending on the acceleration voltage of the electron beam.
Most of the EBM specific algorithms are embedded in the highly parallel lattice Boltzmann framework waLBerla. The metal powder particles are simulated by the also highly parallel physics engine pe.
Within the coupling particles are represented as rigid bodies in the pe and treated as boundaries in the LBM scheme of waLBerla. Both frameworks work on state-of-the-art supercomputers. The EBM application and its implementation are validated by benchmarks where analytical solutions are common knowledge. Moreover, the simulation results are compared to experimental data with respect to quality of the product in order to avoid porosity, and ensure dimensional accuracy. Since the numerical and experimental data are highly concordant the implemented EBM model is suitable to develop new processing strategies in order to improve the quality of the products. The simulations support machine users and developers in order to find an optimal parameter set for specific parts.
Lastly, the accuracy order of the applied free surface boundary condition is examined via the Chapman-Enskog expansion since it has a huge influence on the simulation results. It is established that the original FSLBM boundary condition is just first order accurate for general cases and since the LBM is second order accurate the overall accuracy is reduced by applying FSLBM. In order to overcome this deficiency an improved second order FSLBM boundary condition is derived successfully. The importance and correctness of this new FSLBM boundary condition is finally underlined by a thorough validation against analytical
calculations and experiments.

The increasing interest in microfluidic applications in recent years resulted in a constantly growing demand for computational fluid dynamics on micro-/mesoscopic scales. The corresponding methods have evolved from a topic mainly of relevance for fundamental research to a crucial tool for engineering
applications. The physically correct modeling of the boundary interactions is critical on these scales, as the influence of boundary effects increases inversely proportionally with the characteristic length. Mesoscopic particle-based approaches, that is, methods where the fluid is modeled using discrete particles, are able to capture these effects correctly. The applicability of many existing simulation tools, however, is rather limited with respect to systems of complicated shape. Yet such shapes are necessary for
microfluidic devices in order to perform, for example, the continuous sorting of cells. Furthermore, as the computational cost of mesoscopic particle-based methods is high, the simulation tools have to be optimized for modern high-performance computing systems and be able to exploit the parallelism of today's supercomputers.
The aim of this work is to develop methods and algorithms required for a general, yet efficient simulation framework able to handle boundary conditions of complicated shape. The description of the computational domain using unstructured grids, combined with additional obstacles defined by combinations of geometric primitives is suggested. This hybrid approach combines the
universality of unstructured grids constructed by applying conventional meshing tools with the fully analytical description of boundary surfaces. This analytical description aligns conceptionally with the particle-based modeling of fluids and completely avoids the necessity to perform a discretization of the boundary. We devise an efficient and numerically robust algorithm for the
maintenance of neighbor lists in such complicated geometries. Various physically motivated boundary conditions are tightly integrated into this algorithm, allowing for the simulation of fluid flows in complex geometries. A thorough validation using standard test cases is performed to ensure the correct working of the implemented particle models. Several example applications for flows through complicated geometries are presented, such as
the flow around a particle cluster and the flow through a highly porous medium.
Using the unstructured grid as a basis, a fully object-oriented simulation framework is developed, which is easily extendible by separating the physical models from the data handling and parallelization scheme. The efficiency of the proposed parallelization scheme is assessed via benchmark runs on different
machines, highlighting the versatility and quality of the simulation tool.

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.

This thesis is concerned with the solution of contact problems with advanced Coulomb friction in the 3D case using the finite element method. A Lagrange multiplier method
modelling the contact traction is employed and the contact conditions are enforced in a weak sense leading to a surface-to-surface discretization. Here more precisely the dual mortar method is used allowing for a static condensation of the additional variables in the system before solving without loosing the optimality of the solution. The discrete contact inequalities are embedded in the algebraic system using a semi-smooth Newton
method handling all system non-linearities, geometrical, material and contact within one iteration loop.
In contrast to other research in this field, this is the first time, that the dual Lagrange method is combined with constitutive contact laws considering the surface roughness on the micro scale. In normal direction this is achieved via a perturbed Lagrange
method using a regularization of the contact conditions being able to model a transition
from soft to hard contact. In tangential direction the transition from elastic stick over micro slip to macro slip needed for frictional damping is considered. The elastic
stick is modelled again with a perturbed Lagrange method while the micro slip satisfying the Masing rules for arbitrary load path is modelled with the serial-parallel Iwan model.
Thus a perturbed Lagrange dual mortar method for contact problems is derived and studied in detail.
The method is first derived for the quasi-static case using linear finite elements. Then it is extended to quadratic finite elements using a transformation of the basis functions. Also extensions suitable for complex applications including cyclic symmetry and directional blocking or hanging nodes are presented.
In the last part of this work the fully dynamical contact problem with friction is considered. The generalized alpha-method is used for discretization in time and the algebraic system is derived with respect to accelerations. Again, the contact conditions are embedded using a semi-smooth Newton method and the dual Lagrange multiplier can be
condensed from the system enabling the calculation of frictional damping as needed in structural dynamics. Numerical examples including various contact zones, non-linear material laws and
comparisons with measurements show the wide applicability of the derived algorithms.

Rigorous Fourier methods are methods for the rigorous calculation of the scattering of waves at gratings, which are based on Fourier expansions of the field and material distribution in the direction(s) of periodicity; they are of importance for the calculation of optical systems containing diffractive elements – e.g. interferometers. In the direction(s) of periodicity, the time-independent Maxwell equations are projected onto a Fourier basis. The remaining ordinary differential equation defines together with the boundary conditions at the homogeneous medium of incidence and transmission the boundary value problem, the subject of this thesis. Its solution is given by the scattering-matrix (S-matrix) to be calculated.
The fulfillment of the boundary conditions is possible in different ways: The Conventional Differential Method is based on the Shooting Method. Its disadvantage is numerical integration along exponentially increasing functions (anti-evanescent waves). As naturally stable pendant to this, the Direct S-matrix Integration has been developed in the context of this thesis: It integrates the S-matrix directly by a differential equation derived for this. Both methods have O(N³)-complexity, with N the number of harmonics.
Recently, alternative rigorous Fourier methods of merely O(N·ln(N))-complexity based on integral equations are emerging. Improvable points here are the number of iterations for the solution of the integral equation, the memory consumption and a lack of flexibility. One remedy to this developed in the context of this thesis is the S-vector algorithm.
The presented methods are combined with polarization ray tracing as well as the angular spectrum of plane waves outside gratings. The specified examples show the need, applicability and merit of rigorous Fourier methods.

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.