Dipl.-Math. Patrick Schön

Patrick Schön

Department for Applied Mathematics
Albert-Ludwigs-University Freiburg
Hermann-Herder-Str. 10, Room 207
79104 Freiburg im Breisgau

Tel: 0761 / 203 5647
E-Mail: patrick.schoen@mathematik.uni-freiburg.de
Office hours: Wed 13:00 - 15:00

Research

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.

  • Sören Bartels and Patrick Schön. Local coarsening of simplicial finite element meshes generated by bisections. BIT, 52(3):559--569, 2012. [ bib | DOI | http | .pdf ]

Liquid Crystals

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.

The videos show a switching process in a prototype Zenithally Bistable Device via an applied electric field from the Hybrid Aligned Nematic (HAN) state to the Vertically Aligned Nematic (VAN) state and vice versa.

Monge-Kantorovich Problem

The 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.
  • Sören Bartels and Patrick Schön. Adaptive approximation of the Monge-Kantorovich problem via primal-dual gap estimates. 2016. [ bib | .pdf ]

Teaching

  • Sommersemester 2014:
      Vorlesung, Grundlagen der Programmiersprache C
  • Sommersemester 2013:
      Proseminar, Numerik, Prof S. Bartels