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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.

We have analyzed and implemented stable schemes for two-phase flow with species transport. In the first model, the species taken into account is to be interpreted as a density of charges. It interacts with external electrostatic potentials and, which is much more exciting, with the fluids. The second model presented here is similar in its form but takes non-matched mass densities into account (instead of species transport). Both models cover a set of fluid dynamic effects. For both models, we give a definition of the quantities and formulate assumptions for the physical or phenomenological parameters, such as viscosity, permeability and the interface thickness.
With the intention of a rigorous numerical analysis for model 1, a specific discretization scheme in space and time is suggested. Based on this scheme, we show the discrete version of the energy estimate for both models and prove the existence of discrete solutions for the electrowetting model. Together with sufficient regularity in time, this enables us to show weak convergence for subsequences. The limits of these subsequences can be identified as solutions of the continuous weak formulation. Basic numerical analysis for the case of non-matched densities is presented. We formulate a fully discrete scheme for model 2 and prove its stability by showing a discrete energy estimate.
Some selected simulations based on both models are presented. Both models have their numerical pitfalls, so we allow for basic tests to assure the validity of the code. The inhouse code EconDrop is described. The software package allows for a large variety of numerical problems, e.g. Navier-Stokes equations, generalized Poisson's equation and advection-diffusion equations. It offers adaptivity in both time and space.

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.

Based on Onsager's variational principle, the motion of superparamagnetic nanoparticles – which are suspended in an incompressible carrier fluid and subjected to an external magnetic field – is described by systems of partial differential quations. Various modeling assumptions lead to three different systems denoted by "model GW", "model W" and "model B". When proposed in our joint work (2019), "model GW" was – to the best of our knowledge – the first one to include evolution equations for the magnetization field and for the magnetic particle density – all of them are nonlinearly coupled to the magnetostatic and the Navier-Stokes equations. The other two models use algebraic equations to determine the magnetization based on the linearized Langevin formula and are derived by us for the purpose of comparison. In case of "model W", we assume that the suspension yields a single-phase flow with negligible mass of magnetic nanoparticles. The other model is derived under the assumption that fluid particles and magnetic particles set up a two-phase fluid. It shows similarities to the model from Himmelsbach et al. (2017). All three models follow a two-domain approach – considering the magnetic field on a possibly larger domain compared to the fluid domain – of which we expect higher accuracy in determining the total magnetic field. The case when the two domains coincide requires some slight changes which are discussed when needed.
In contrast to other works in the mathematical literature, the boundary conditions of the magnetization equation in "model GW" are motivated by physical arguments only, not by practical aspects with respect to mathematical analysis. They entail H(div,curl)-regularity of the magnetic quantities – magnetization and (total) magnetic field – but possibly not H1-regularity. To establish existence of solutions to this model, the absence of H1-regularity is compensated by an intricate approximation procedure. For this, we give a meaning to the Kelvin force (m*nabla)h in the distributional sense. Existence of distributional global-in-time solutions is guaranteed under appropriate assumptions. The latter include nonlinear diffusion to be used in the evolution equation for the magnetic particles' density and the restriction to the two-dimensional setting. An existence result of global-in-time weak solutions to a regularized model is presented also in the three-dimensional setting.
To each of the three models, we propose an unconditionally energy stable finite element scheme. The discretization of "model GW" has already been published in our joint work (2019). Here, non-conforming finite elements are used to approximate the magnetization – similar as in a paper of Nochetto et al. (2015). For the first time, however, the second order differential operators nabla div and curl curl in the magnetization equation have been discretized by introducing the operators divh and curlh – discrete versions of div and curl. The latter are defined by duality and yield H1-conforming approximations of the divergence and curl of a vector field. By means of Schaefer's fixed point theorem, existence of discrete solutions is guaranteed for all three schemes. The existence results are independent of the discretization parameters.
The thesis concludes with simulations that serve as a proof of concept for the models and their numerical schemes. The three models are compared in the case of linear diffusion and the case of nonlinear diffusion which has been used in the analysis part of this thesis. For simplicity, most simulations are performed under the assumption that the domain of the magnetic field coincides with the fluid domain. Using "model W" – for practical reasons – the effects of multiple different external magnetic fields are examined as well as the impact of using a strictly larger domain for the magnetic field.