Current research areas
My research portfolio has one established mathematical center and three connected directions. Geometry-aware finite-element methods form the core. Constrained fields extend this work to the geometry of the unknown, geometric evolution provides demanding applications, and computational abstractions make the methods implementable and reusable.
A. Geometry-aware finite-element methods
How does the discrete representation of a domain enter a finite-element method, and which geometric information is actually required for accuracy and stability?
Surface finite-element methods approximate both a PDE and the curved surface on which it is posed. Classical estimates treat errors in position, normals, and curvature separately. In a variational formulation, however, these quantities occur in particular combinations. Leading errors can cancel, so the consistency error may converge faster than pointwise estimates suggest.
Established work
My work on dune-curvedgrid makes higher-order surface parametrizations available in DUNE. With Hanne Hardering, I analyzed tangential vector and tensor fields and surface Stokes discretizations on approximated surfaces. Our recent preprint on parity-dependent geometric consistency errors (see Hardering et al. 2026) explains why even-order geometry approximations can show unexpected superconvergence on suitably structured meshes.
Current themes
I am investigating stabilized reconstructions of mean curvature and the shape operator, including their higher-order variants. A related question is how these reconstructions compare with Hellan–Herrmann–Johnson (HHJ) and Regge finite elements. These spaces encode different continuity properties of symmetric tensor fields and may provide compatible representations of the shape operator, its cofactor, and mean or Gaussian curvature.
Questions ahead
Smooth geometric quantities satisfy identities such as the Gauss–Codazzi equations, but separately reconstructed discrete quantities generally do not. I am exploring whether useful discrete compatibility conditions can be identified and whether they improve approximation or robustness. This is a research direction, not yet an established theory.
Selected work: surface Stokes analysis, tangential errors, odd behaviour of even geometries, and dune-curvedgrid. See also Area A projects.
B. Finite elements for geometrically constrained fields
Given several constraints, which should be encoded exactly in the finite-element space and which should be imposed approximately?
Surface liquid crystals provide a concrete motivation. Their vector or tensor order parameters may need to be tangential, symmetric, trace-free, of fixed length, or invariant under a head-tail symmetry. The broader numerical question applies to many PDEs whose unknowns take values in geometry-dependent spaces.
Established work
Earlier computational studies compared representations of orientational order on surfaces and extended them to symmetric tensor fields. Later analysis separated tangential and normal approximation errors and showed that optimal tangential estimates for penalty methods do not always require a higher-order normal. Current work on geometric surface finite elements combines an approximated surface domain with a manifold-valued range.
Current themes
In the DFG project Symmetry, length, and tangential constraints, Hanne Hardering, Valentin Neumann, and I compare complementary discretizations of tangential unit-vector fields. One approach represents tangentiality exactly and penalizes unit length. The other represents unit length exactly and penalizes tangentiality. We are also studying exact tangential constructions based on geometry-dependent transformations between elements.
The next concrete step is to extend these ideas from vectors to tensor-valued fields and to understand which transformations preserve tangentiality, symmetry, and trace conditions. This connects directly to HHJ, Regge, and Arnold–Winther elements.
Questions ahead
Tangential fields are sections of a tangent bundle, since their admissible values change with position. Finite elements for sections of more general fibre bundles are a longer-term perspective. They would require a consistent treatment of both the approximated base surface and the changing target spaces.
Selected work: tangential errors, surface Stokes analysis, and the DFG constrained-fields project. See also Area B projects.
C. Structure-preserving geometric evolution
How can a numerical scheme preserve the geometric and variational structures of an evolution equation while its domain changes?
Geometric evolution is where domain approximation and constrained geometric quantities meet. Curvature, normals, and the shape operator may drive the motion, while the continuous equation may dissipate an energy or preserve volume. The discrete method should reproduce the relevant structure without hiding errors in the geometry approximation.
Established work
With Harald Garcke, Robert Nürnberg, and Ganghui Zhang, I developed higher-order isoparametric methods for mean-curvature flow and surface diffusion. The schemes retain energy stability, and the structure-preserving surface-diffusion method conserves enclosed volume. Our work on surface Minkowski tensors provides intrinsic, tensor-valued measures of rotational symmetry for shapes on curved surfaces.
Current themes
Curvature energies and flows. I am studying the gradient flow of the squared Gaussian-curvature energy. Its driving force involves the shape operator, its cofactor, Gaussian curvature, and derivatives of these quantities. This motivates mixed formulations that avoid direct approximation of very high derivatives and provides a demanding test for the curvature discretizations in Area A.
Minkowski measures. Surface Minkowski tensors lead to a separate exploratory question: can a flow optimize or control a global, tensor-valued shape measure? Numerical experiments and analysis for simpler settings are under way. This branch is related to structure preservation, but it does not yet form a unified theory with curvature-energy flows.
Questions ahead
Longer-term questions include geometric evolution under nonclassical global constraints and the coupling of evolving geometry to constrained surface fields.
Selected work: isoparametric mean-curvature flow and surface diffusion, surface Minkowski tensors, and SurfaceMinkowski.jl. See also Area C projects.
D. Computational abstractions for advanced finite elements
Which mathematical abstractions allow advanced finite-element methods to be expressed generically, efficiently, and in a form that can be reused?
Software is not a separate final step in this research. The representation of a curved geometry or a tensor-valued finite element determines which methods can be formulated, compared, and maintained in practice. My long-term work as a core developer in DUNE and AMDiS provides the computational base for the other three areas.
Established work
I rewrote AMDiS as a DUNE module and contribute to DUNE’s core and extension modules. My work includes curved-grid representations, composed finite-element bases with tree-structured indices, periodic grids, mesh comparison, file formats, and tensor data structures. Several of these components have grown directly from research questions or supervised theses.
Current themes
Current development concerns higher derivatives of discrete geometries, tensor-valued range and derivative types, and a general interface for transformed finite elements. These capabilities are needed for Piola transformations on curved cells and for implementations of HHJ, Regge, Arnold–Winther, and related elements. I am also interested in generic finite-element generation and in parallel indexing for composed spaces.
Questions ahead
A framework for bundle-valued finite elements would need representations of fibres, distances, exponential and logarithmic maps or retractions, parallel transport, and their derivatives. The mathematical problem belongs to Area B. The task here is to find computational abstractions that express it without tying the implementation to one particular bundle or PDE.
Selected work: concepts for composed finite-element bases, DUNE and AMDiS, and the geometry and numerical infrastructure. See also Area D projects.
The areas describe the scientific portfolio. For active work, bounded thesis questions, and exploratory ideas, continue to Projects & thesis topics →