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.
Research areas
The four areas below describe the stable themes that connect my work. The detailed research areas explain the established results, current questions, and links between them.
Geometry-aware finite-element methods: Accurate representations of curved domains, metric quantities, normals, curvature, and the shape operator. Read more
Finite elements for geometrically constrained fields: Finite-element spaces for tangential vector and tensor fields with interacting geometric constraints. Read more
Structure-preserving geometric evolution: Geometric flows that retain relevant energy laws, conservation properties, and shape measures. Read more
Computational abstractions for advanced finite elements: Reusable abstractions in DUNE and AMDiS for advanced geometry and finite-element methods. Read more
Current projects
Curvature reconstruction
Stabilized approximations of mean curvature and the shape operator, including higher-order variants and compatible tensor spaces.
Constrained surface fields
Complementary discretizations of tangential vector and tensor fields with symmetry, length, and tangential constraints.
Geometric evolution
Mixed formulations for curvature-driven flows and exploratory evolution controlled by tensor-valued shape measures.
Advanced finite-element infrastructure
Higher geometry derivatives, tensor-valued data, and transformed finite elements in DUNE.
How I work
I combine numerical analysis, the design of finite-element spaces, computational experiments, and reusable scientific software. Implementation often exposes missing assumptions or unexpected numerical effects. The parity-dependent geometric errors found in our surface finite-element experiments are one example. Analysis can explain these effects, and shared software makes improved methods available for further research.
- 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.