 Numerical Analysis Groups

Home    Members    Teaching    Preprints    Publications    Projects    Links ## Christian Power

Mathematisches Institut
Universität Tübingen
Auf der Morgenstelle 10
D-72076 Tübingen

Room 3P16, building C (site map)

phone: +49 7071 29 72934
fax: +49 7071 29 4322
mail: power (at) na.uni-tuebingen.de #### Interests

• Maxmimum norm estimates for ESFEM and time discretization
• Quasilinear parabolic PDE on moving sufaces
• Evolving surface finite element method (ESFEM)
• Arbitrary Lagrange Eulerian maps (ALE) for ESFEM
• Numerical Analysis of the mean curvature flow

#### Numerical experiments

The video above is a finite element solution for $(\vec X, u)$, which solves a surface partial differential equation of the following type $\partial^{\vec X(t)}_t u + div_{\vec X(t)}(\vec v) u - \Delta_{\vec X(t)} u = f,$ $\vec v - \alpha \Delta_{\vec X(t)} \vec v = g(u) \vec n_{\vec X(t)},$ where $\vec X \colon M\times [0,T] \to \mathbb{R}^3$ describes the evolution of the inital surface $M$, with velocity $\vec v(\vec X,t)=\partial_t \vec X$, unit normal $n_{\vec X(t)}$, manifold time derivative $\partial^{\vec X(t)}_t u = \partial_t u(\vec X(t),t)$, manifold divergence $div_{\vec X(t)}$, Laplace-Beltrami operator $\Delta_{\vec X(t)}$, and where $u(t)\colon X(M,t)\to \mathbb{R}$ describes a chemical concentration, which may promote or inhibit the growth of the surface. Hence, the resulting surface deformates in a complicated fashion.

Snapshots from the following two videos have been used in . The left-hand side works with the regularized velocity law above. The right-hand side is regularized with a mean curvature term.

Source code is available on Github. Documentation is available here.

#### Preprints & Publications

1. Balázs Kovács, Buyang Li, Christian Lubich, Christian Power, Convergence of finite elements on a solution-driven evolving surface, July 2016.
2. Balázs Kovacs and Christian Power, Maximum norm stability and error estimates for the evolving surface finite element method, preprint, Septembre 2015.
3. Balázs Kovacs and Christian Power, Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces, revised preprint, March 2015. To appear in NMPDE.
4. Balázs Kovacs and Christian Power, Higher-oder time discretizations with ALE finite elements for parabolic problems on evolving surfaces, revised preprint, February 2015.
5. Christian Power, Der Trajektorienraum des Gradientenvektorfeldes einer Morse-Smale-Funktion auf einer riemannschen Mannigfaltigkeit, Diploma thesis, Mai 2012

#### Teaching & Administration

• winter term 2016/17: teaching assistant for Numerik I (lecture, Prof. Lubich)
• summer term 2016: teaching assistant for Nonlinear partial differential equations (seminar, Prof. Lubich)
• summer term 2016: teaching assistant for Numerik IV (lecture, Prof. Lubich)
• winter term 2015/16: teaching assistant for Analysis I (lecture, Prof. Heinzl)
• sommer term 2015: teaching assistant for Numerik II (lecture, Prof. Lubich)
• winter term 2014/15: teaching assistant for Numerik I (lecture, Prof. Lubich)
• summer term 2014: teaching assistant for Analysis II (lecture, Prof. Lubich)
• winter term 2013/14: teaching assistant for Analysis I (lecture, Prof. Lubich)
• summer term 2013: teaching assistant for Algorithmen der numerischen Mathematik (lecture, Dr. Mansour)
• winter term 2012/2013: system administration for the mathematical institut
• summer term 2012:  system administration for the mathematical institut
• winter term 2011/2012: teaching assistant for Lineare Algebra I (lecture, Prof. Batyrev)
• summer term 2011: teaching assistant for Lineare Algebra II (lecture, Prof. Gubler)
• winter term 2010/2011: teaching assistant for Lineare Algebra I (lecture, Prof. Gubler)