Student projects and theses

I supervise Bachelor's and Master's theses as well as smaller research and programming projects in numerical mathematics and scientific computing. Many projects are connected to current research in finite element methods, geometric PDEs, numerical algorithms, and scientific software.

The topics below are starting points rather than fixed assignments. Their precise scope can usually be adapted to the type of project, your mathematical and programming background, and your interests.

Projects differ in character: some focus primarily on mathematical analysis, others on computational experiments and implementation, while others start from a mathematical model or application. The descriptions below indicate the expected emphasis and useful background.

If you are interested in a topic, please contact me. We can discuss the precise question and whether it is suitable as a Bachelor's thesis, Master's thesis, or research project.

Bachelor's projects

B1: Curvature of discrete and higher-order curves

Focus: Computation / Analysis
Useful background: PR20, Numerical Mathematics

How can the curvature of a smooth curve be recovered when only a piecewise linear or polynomial approximation is available? Different local curvature approximations are implemented and compared on curves with known curvature, and numerical experiments are used to investigate their accuracy and convergence under mesh refinement.

The project can be adapted towards either computational experiments or a more detailed investigation of the approximation error. Basic concepts from differential geometry can be introduced as part of the project.

B2: Characterizing planar shapes with Minkowski tensors

Focus: Computation / Geometry
Useful background: PR20, Linear Algebra

How can we numerically distinguish a regular triangle from a stretched or perturbed one? Minkowski tensors encode information about the shape and anisotropy of an object in tensor-valued geometric quantities. In this project, such tensors and their eigenvalues and eigenvectors are computed for planar shapes and used to characterize their symmetries and preferred directions.

Numerical experiments investigate invariance under translations and rotations as well as robustness with respect to refinement and perturbations of the shape.

B3: Normal reconstruction on triangulated surfaces

Focus: Computation / Analysis
Useful background: PR20, Numerical Mathematics

On a piecewise flat approximation of a curved surface, normal vectors are constant on each triangle and discontinuous across edges. This project investigates methods for reconstructing a continuous and more accurate normal vector field, for example by averaging neighbouring element normals at the vertices.

Different weighting strategies are compared numerically and their convergence under mesh refinement is studied. The project can be extended to more advanced reconstruction techniques such as polynomial patch fitting or finite-element projection.

B4: Symmetries in refined triangulations

Focus: Analysis / Computation
Useful background: PR20, Numerical Mathematics

Structured refinement of a triangular grid produces pairs of triangles that are mirror images of each other. Such local symmetries can cause leading interpolation errors to cancel and thereby produce unexpectedly high convergence rates.

The project investigates how these symmetric triangle pairs arise and aims to characterize or prove their occurrence for selected refinement procedures. A computational variant can compare different refinement strategies and measure the resulting interpolation errors.

A related project can investigate deliberately constructed mesh sequences without these symmetries. The two questions are also suitable as coordinated projects for two students.

B5: Grid generation with Gmsh and DUNE

Focus: Scientific Software / Computation
Useful background: PR20; C++ programming
Programming: C++

Gmsh is a widely used mesh-generation library, while DUNE provides C++ abstractions for computational grids and finite-element methods. This project investigates how information generated by Gmsh can be transferred more directly into DUNE.

Possible directions include reading periodicity information from Gmsh mesh files, reconstructing the corresponding topological connections in DUNE, or using the Gmsh C++ API directly to construct DUNE grids without an intermediate mesh file. The precise direction can be chosen according to the scope of the project and the student's programming experience.

Master's projects

M1: Stabilized finite-element approximation of curvature

Focus: Computation / Numerical Analysis
Useful background: Finite Element Methods, Numerical PDEs

Curvature quantities computed directly from discrete surfaces often converge less accurately than desired. Stabilized finite-element reconstructions provide one approach to improving these approximations.

In this project, stabilized methods for the mean-curvature vector and the shape operator are implemented and systematically compared with their non-stabilized counterparts. Numerical experiments investigate accuracy and convergence as well as properties and conditioning of the resulting linear systems.

M2: Analysis of stabilized finite-element curvature approximation

Focus: Mathematical Analysis
Useful background: Finite Element Methods, Numerical PDEs, Functional Analysis

This project studies the mathematical foundation of stabilized finite-element methods for approximating curvature. Starting from existing results for the mean-curvature vector, the goal is to derive discretization-error estimates on piecewise flat approximations of curves and surfaces and to investigate how the analysis extends to the approximation of the shape operator.

This topic is closely related to the computational curvature project above and could also form part of a coordinated project combining theoretical and numerical investigations.

M3: Surface Stokes equations in stream-function formulation

Focus: Analysis / Finite Element Methods
Useful background: Finite Element Methods, PDEs, Functional Analysis

The surface Stokes equations describe viscous flow constrained to a curved surface. In a stream-function formulation, the tangential velocity is represented through derivatives of a scalar field, leading to a discretization that differs substantially from the usual velocity-pressure formulation.

The project investigates the discretization error of a surface finite-element approximation of this formulation, including the reconstruction of velocity and pressure. Particular attention is paid to errors caused by the approximation of the surface geometry and the curvature quantities entering the method.

M4: Discretization of a tangential vector Poisson equation

Focus: Computation / Numerical Analysis
Useful background: Finite Element Methods, Numerical PDEs

A vector field defined on a surface may be required to remain tangential at every point. There are several fundamentally different ways to represent this constraint numerically: intrinsically, through penalty terms, or using Lagrange multipliers.

This project implements and compares these approaches for a vector Poisson problem on a curved surface. Numerical experiments investigate approximation errors, tangentiality, and the influence of parameters such as penalty strengths and the choice of Lagrange-multiplier space.

M5: Higher derivatives of DUNE geometries

Focus: Scientific Software / Numerical Geometry
Useful background: Scientific Programming, Finite Element Methods or Differential Geometry
Programming: C++

Curved computational grids are represented by element-wise geometry mappings. Their first derivatives provide tangent vectors and normals, while second and higher derivatives are needed for quantities such as curvature and for advanced finite-element transformations.

The project develops an extension of the DUNE geometry interface for higher derivatives, implements it for selected geometry types, and evaluates the design by computing geometric quantities such as the first and second fundamental forms. This project combines numerical geometry with the design of reusable scientific-software interfaces.

Further project ideas

Additional projects can be developed around current research questions. Examples include:

Minkowski spectra for shape recognition — investigate whether tensor-valued geometric descriptors can be used as features for classifying shapes, and compare them with image-based approaches.
Rational interpolation on triangular domains — investigate extensions of Floater–Hormann-type rational interpolation from intervals to triangles and study their approximation and stability properties.
Mesh sequences with and without local symmetries — computationally construct and compare mesh families in which symmetry-induced cancellation effects are present or absent.

Other topics in finite element methods, geometric PDEs, numerical algorithms, and scientific software may also be possible. Please contact me if you are interested in developing a project related to these areas.

Theses of the past years

So 2025: Master's thesis: SynCast - Enhancing Weather Forecast Precision by AI-based Multi-Model Data Fusion verified by Observations
Wi 2023/24: Master's thesis: Code Generation for Finite Element Simulation: Interfacing AMDiS from Python
Wi 2023/24, Master's thesis: Overlapping Domain Decomposition Methods for the Efficient Solution of Partial Differential Equations
Wi 2023/24, CMS Research Project: Parallelization via Graph Coloring in FEM
Wi 2023/24, Research Project: Cahn-Hilliard Equation in AMDiS: Analysis and Comparison
So 2022, Master's thesis: Periodic Boundary Conditions on non-periodic Meshes for Systems of PDEs: Methods, Implementation, Examples and Analysis
Wi 2021/22, Master's thesis: Collective behaviour of active brownian particles by active VPFC modeling
Wi 2021/22, Master's thesis: Data exchange between independently refined grids and its application in multi-phase-field models
Wi 2021/22, Master's thesis: H2 (non)conforming Finite Elements for the DUNE framework
Wi 2021/22, Master's thesis: Machine Learning prediction models for functionality and performance parameters of semiconductor chips based on manufacturing datasets
So 2021, Research project: Defect-tracking in Active Smectics
So 2021, Master's thesis: Modellvergleiche des maschinellen Lernens zur automatischen Applikation von Sitzpersonalisierungsfunktionen
Wi 2019/20, Research project: Restoration of Binary Images Using the Cahn-Hilliard Equation
So 2019, Modelling Seminar: Fluid particle dynamics