Current research projects
These projects turn the broader research areas into concrete analytical, computational, and software questions. Ongoing projects are active work. Exploratory questions describe longer-term directions that still need further preparation.
A. Geometry-aware finite-element methods
ONGOING
Stabilized curvature and shape-operator reconstruction
Direct differentiation of an approximated surface loses accuracy with each derivative. This project studies finite-element reconstructions of mean curvature and the shape operator that add edge-jump stabilization. Initial experiments show improved convergence. Current work concerns analysis, higher-order variants, conditioning, and comparison with HHJ-based reconstruction.
HHJ and Regge spaces encode different continuity properties of symmetric tensor fields. They may provide compatible representations of the shape operator, its cofactor, and mean or Gaussian curvature.
Related work: odd behaviour of even geometries · dune-curvedgrid
EXPLORATORY
Compatible discrete geometry
HHJ and Regge spaces provide complementary representations of the shape operator and its cofactor. A larger question is whether these quantities can be approximated so that a meaningful discrete analogue of Gauss-Codazzi compatibility is retained. This direction needs further analytical preparation before it becomes a defined project.
B. Finite elements for geometrically constrained fields
ONGOING
Symmetry, length, and tangential constraints
This joint project with Hanne Hardering and doctoral researcher Valentin Neumann studies vector and tensor fields with several interacting constraints. Current work compares exact tangentiality with penalized unit length against exact unit length with penalized tangentiality. We are also studying exact tangential constructions based on geometry-dependent transformations between elements.
The aim is to understand approximation, stability, penalty scaling, and the effect of the discrete surface geometry. The next step is to extend these ideas from vectors to tensor-valued fields and to determine which transformations preserve tangentiality, symmetry, and trace conditions. This connects directly to HHJ, Regge, and Arnold-Winther elements.
Related work: tangential errors of tensor surface finite elements · Area B overview
EXPLORATORY
Finite elements for bundle-valued fields
A tangential field takes values in spaces that vary with position. Extending geometric finite elements to more general fibre bundles would require new definitions of interpolation, lifting, and conformity when both the domain and target fibres are approximated. This is a longer-term mathematical and computational direction.
C. Structure-preserving geometric evolution
ONGOING
Squared Gaussian-curvature flow
The gradient flow of the squared Gaussian-curvature energy involves the shape operator, its cofactor, Gaussian curvature, and their derivatives. The current project develops mixed formulations that avoid a direct discretization of very high derivatives and compares alternative curvature approximations. It provides a demanding test for the curvature discretizations in Area A and is still in the research phase.
Related work: geometry-aware finite-element methods · structure-preserving evolution
ONGOING
Evolution controlled by surface Minkowski measures
Surface Minkowski tensors quantify rotational symmetry through global, tensor-valued shape measures. Current analysis and experiments ask whether a curve on a surface can evolve toward a prescribed symmetry or optimize such a measure. Numerical experiments and analysis for simpler settings are under way. This branch is separate from the curvature-energy project and remains exploratory.
Related work: surface Minkowski tensors · SurfaceMinkowski.jl
EXPLORATORY
Coupled field and geometry evolution
If a surface evolves, its tangent spaces and the constraints on attached fields evolve as well. A future direction is to couple structure-preserving surface motion with constrained vector or tensor fields and determine which continuous identities should survive discretization.
D. Computational abstractions for advanced finite elements
ONGOING
Higher-order geometry and transformed finite elements in DUNE
Curvature calculations and Piola-transformed elements on curved cells require derivatives that are not yet part of the general DUNE geometry interface. Current work develops suitable interfaces for higher derivatives, tensor-valued ranges, and transformed elements. These capabilities support HHJ, Regge, Arnold-Winther, and related finite elements. Related directions include generic finite-element generation and parallel indexing for composed spaces.
Related software: DUNE and AMDiS · dune-curvedgrid · dune-tensor
EXPLORATORY
Interoperable descriptions of advanced finite elements
Libraries such as DUNE, FEniCSx, NGSolve, deal.II, and Symfem represent advanced elements in different ways. A longer-term question is whether element definitions or generated basis data can be exchanged without losing the transformations and tensor structure required by an implementation.
Student projects
I supervise Bachelor’s and Master’s theses connected to these research areas. The current list of available topics is maintained only on the TU Dresden thesis page.