Abteilung für Angewandte Mathematik
79104 Freiburg im Breisgau
Efficient adaptive mesh refinement
For time-dependent partial differential equations coarsening is an important ingredient to develop efficient adaptive mesh refinement strategies. When interfaces or singularities of solutions advance in time or during an iterative method the refined region of the underlying grid of the Finite Element space should follow the interface or singularity. These phenomena occur for example in phase field models.
In order to develop efficient parallel and adaptive mesh decomposition algorithms that allow for arbitrary repartitioning it is important, that the coarsening strategy does not depend on an explicit knowledge of the refinement history. A local coarsening strategy allows the removal of single nodes that are created via compatible bisection of neighboring simplices. The refinement history is stored implicitly in the ordering of grid elements.
Liquid crystals are anisotropic liquids with many physically interesting features. A nematic LC – the simplest form of LCs – consists of elongated or rod-like molecules that exhibit an orientational order which in turn leads to optical properties. Nowadays, LCs find a use in many smartphones, flatscreens, ebooks and other technical devices. One of their striking characteristics is their response to externally applied electric fields.
Under the influence of electric fields, splay bend structures may occur, which allows for switching between two stable states in bistable devices. These Phenomena can be modelled via the Landau-de Gennes theory for nematic LCs.
Monge-Kantorovich ProblemThe Monge-Kantorovich Problem consists in finding the optimal transport costs between two measures. It leads to a nonsmooth convex minization problem with applications in economics, image processing and data analysis.