Research overview
My research lies at the intersection of numerical analysis, geometric partial differential equations, and scientific computing. I study how the geometry of a computational domain, the constraints on unknown fields, and the structure of an evolution equation can be represented and preserved in finite-element discretizations.
Three guiding questions
Geometry enters a numerical method at several levels at once. A surface is only approximately represented by a mesh. Vector and tensor fields may have to remain tangential, symmetric, or of fixed length. If the surface evolves, the discretization should also respect variational identities, energy laws, and conserved geometric measures.
01
How should the geometry of the domain be discretized?
I investigate which approximations of position, metric, normal vectors, curvature, and the shape operator are needed for accurate and stable surface finite-element methods.
02
How should the geometry of the unknown fields be represented?
I study finite-element spaces for tangential vector and tensor fields and ask which constraints should be built into the space and which should be imposed by penalties or Lagrange multipliers.
03
How can these structures be preserved when the geometry evolves?
I develop and test discretizations of geometric flows that retain properties such as energy decay, volume conservation, or prescribed geometric measures.
One connected problem
The three questions cannot be separated completely. The admissible values of a tangential field depend on the geometry of its domain. Curvature may enter a surface PDE as a coefficient, or it may drive the motion of the surface itself. When the geometry is approximated, these connections affect consistency, stability, and the qualitative behavior of the computed solution.
Domain
Higher-order surface approximation makes geometric information available. The central analytical question is which combinations of metric, normal, and curvature errors actually enter a weak formulation.
Fields
Tangentiality, symmetry, trace, length, and manifold constraints can be represented exactly or approximately. Their interaction determines approximation quality, conditioning, and computational cost.
Evolution
Normals, curvature, and geometric measures become part of the unknown when a surface moves. Their discrete representation influences energy laws, conservation properties, and mesh quality.
Computation as a transversal theme
Advanced finite-element methods need software concepts that go beyond scalar polynomial elements. My work in the DUNE and AMDiS ecosystems includes curved grids, composed finite-element bases, vector-, matrix-, and tensor-valued data, and transformed elements on curved cells. These abstractions make it possible to compare methods in a common framework and to reuse the results beyond one application.
This interaction also works in the other direction. Implementing a method often exposes missing assumptions or unexpected numerical effects. The parity-dependent geometric errors found in our surface finite-element experiments are one example. Computational experiments can reveal a phenomenon, analysis can explain it, and software can make the improved method available to others.
Research approach
- Numerical analysis and approximation theory: identify the geometric consistency and stability properties that control an error estimate.
- Design of finite-element spaces: choose continuity, transformations, and exact or approximate constraint enforcement to match the problem.
- Computational experiments: test assumptions, compare formulations, and discover numerical phenomena that are not visible in pointwise estimates alone.
- Reusable scientific software: express mathematical structure through documented and tested abstractions that support further research.
Continue through the research
The next level describes the scientific themes, established work, and current questions. Continue to the four research areas →
Looking for something specific? Go directly to open thesis topics, selected publications, or scientific software.