Ein Überblick über Forschungsgebiet, Werkzeuge und Methoden

Former projects

Numerical simulation of two-phase flows with incompressible Navier-Stokes equations and free capillary interphase

Project C2 in the contect of the DFG-research-group Nonlinear Partial Differential Equations

This work attends to the flow of two immiscible, incompressible fluids in two dimensions. The main topic is the kinesic behaviour of the interphase. Velocity and pressure are described by the Navier-Stokes equations and are subject to the capillary boundary condition on the interphase. The work covers modelling of the problem, mathematical treatment of the formulations, discretisation by a finite-element method and the numerical implementation of a program that computes accordingly activities and plots graphical simulations.

Numerical Methods for Incompressible Flow with many Capillary Boundaries

Project C2 in the contect of the DFG-research-group Nonlinear Partial Differential Equations

The goal is the computation of incompressible flow with free capillary boundaries where topological changes of the fluid-domain are of particular interest. For this Level-set methods are an adequate technique. One focus of this project is the development of stable numerical methods for the Navier-Stokes-equations with capillary boundaries which are not fitted to the underlying finite element mesh. Another point is the problem of mass conservation.

The Cahn-Hilliard equation on moving surfaces

Project A1 within the DFG-Forschergruppe Nonlinear Partial Differential Equations

The chemical process of dealloying and the formation of nanoporisity can possibly be described by a model involving a Cahn-Hiliard type equation on a surface moving with a coupled velocity law. Parametric finite element methods are used for the discretization of this problem. In addition, analytical and numerical results for a simplified version of the equations are derived.

Numerical Ricci flow

Ricci flow plays an important role in the proof of the Poincaré conjecture. In contrast to its outstanding relevance previous results on the numerical Ricci flow are however restricted to special cases. In this project we plan to develop a numerical algorithm which principally works for higher dimensions. The basis for our research is a weak form of the Ricci curvature.

Rotating Drops

Equilibrium figures of isolated, viscous fluids are rotating rigid bodies, kept together by surface tension. The simulations were carried out by an (iso-)parametric finite element method.

Computing eigenvalues with higher order Finite Elements

The discretization of the Helmholts-equation with the Finite Element method leads to a generalized eigenvalue-problem. In order to compute many eigenvalues with high accuracy we use higher order Finite Elements due to there approximation-properties. In this project with the Institut für Kernphysik at the TU Darmstadt we compare the computed eigenvalues with experimental results.

Anisotropic mean curvature flow in higher codimension

Project A1 within the DFG-research group Nonlinear Partial Differential Equations

Anisotropic mean curvature flow has been widely studied, both from a numerical and analytical point of view. Until now research mostly concentrated on the study of the motion of hypersurfaces. The goal of this projekt is to study the problem in higher codimension.

We consider the anisotropic mean curvature flow for parametric curves and two dimensional surfaces in R^n for arbitrarily large n, and study its finite element approximation.