Surface Vector and Tensor Fields

© Simon Praetorius

The discretization of vector and tensor valued PDEs on curved hypersurfaces introduced methodical and analytical challenges compared to scalar valued equations. A reason is the tight coupling between the tensor field components and geometric properties of the surface, like its local curvature. Representation of tensor fields is one question that plays a role in the numerical implementation. Either we consider the surface as embedded in a surrounding space and represent the tensor fields as part of this embedding, or we seek for a more intrinsic description of the surface and the fields.

While extrinsic models often allow a simple construction of a convent numerical method, those might involve additional stabilization terms due to its degeneracy. The resulting descriptions are often non-conforming leading to a more involved numerical analysis. Intrinsic methods, on the other hand, often show a very nice theoretical basis, but are often harder to implement in code.

Research Focus

The research project focusses on the development of numerical schemes for vector and tensor fields on surfaces and to compare these methods to each other. Targeting stationary or evolving surfaces raises the question of which method to prefer in which context. All approaches have some restrictions and some fields where they are advantages.

These abstract methods are then applied to applications involving vector fields, like surface fluid flow or polarization fields for surface activity, and tensor fields, like liquid crystal monomers attached to fluid interfaces or surface stresses in fluid mechanics.

Student projects and thesis topics

• Reconstruction of higher-order normal fields on discretized surfaces.
• Analysis of the discrete surfaces of even polynomial order.
• Discretization of the surface Stokes equation with surface finite elements and its numerical analysis.
• Development of discretization methods for subspaces of surface tensor fields, like symmetric tensors, trace-less tensors, Q-tensors, or nQ-tensors.
• Intrinsic surface finite element methods for tangential vector fields.

Publications

• H. Hardering and S. Praetorius. Tangential errors of tensor surface finite elements. (in preparation)
• P. Brandner, T. Jankuhn, S. Praetorius, A. Reusken, and A. Voigt. Finite element discretization methods for velocity-pressure and stream function formulations of surface Stokes equations. (submitted) [arxiv]
• S. Praetorius and F. Stenger. Dune-curvedgrid - a dune module for surface parametrization. (submitted) [arxiv]
• S. Praetorius, A. Voigt, R. Witkowski, and H. Löwen. Active crystals on a sphere. Phys. Rev. E, 97:052615, May 2018. [doi] [arxiv]
• I. Nitschke, M. Nestler, S. Praetorius, H. Löwen, and A. Voigt. Nematic liquid crystals on curved surfaces - a thin film limit. Proc. Math. Phys. Eng. Sci. 474(2214):20170686, June 2018. [doi] [arxiv]
• M. Nestler, I. Nitschke, S. Praetorius, and A. Voigt. Orientational order on surfaces - the coupling of topology, geometry, and dynamics. J. Nonlinear Sci., July 2017. [doi] [arxiv]
• K. Padberg-Gehle, S. Reuther, S. Praetorius, and A. Voigt. Transfer Operator-Based Extraction of Coherent Features on Surfaces, pages 283-297. Springer International Publishing, June 2017. [doi]