Fachbereich Mathematik 
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Henrik Schumacher

Department of Mathematics
Research Group Approximation and Optimization
Bundesstr. 55 (Geomatikum)
20146 Hamburg
Room 105
phone: +49 40 42838-5156
email: henrik.schumacher (at) uni-hamburg.de

Office hours in the Summer Term 2017:
on request (per email)


On $H^2$-gradient Flows for the Willmore Energy
Henrik Schumacher.
Abstract: We show that the concept of $H^2$-gradient flow for the Willmore energy and other functionals that depend at most quadratically on the second fundamental form is well-defined in the space of immersions of Sobolev class $W^{2,p}$ from a compact, $n$-dimensional manifold into Euclidean space, provided that $p \geq 2$ and $p>n$. We also discuss why this is not true for Sobolev class $H^2=W^{2,2}$. In the case of equality constraints, we provide sufficient conditions for the existence of the projected $H^2$-gradient flow and demonstrate its usability for optimization with several numerical examples.

Variational Convergence of Discrete Minimal Surfaces
Henrik Schumacher and Max Wardetzky.
Abstract: Building on and extending tools from variational analysis, we prove Kuratowski convergence of sets of simplicial area minimizers to minimizers of the smooth Douglas-Plateau problem under simplicial refinement. This convergence is with respect to a topology that is stronger than uniform convergence of both positions and surface normals.

Research Reports and Course Notes :
Oberwolfach Reports
Convergence of Discrete Elastica
Henrik Schumacher, Sebastian Scholtes, and Max Wardetzky. Oberwolfach Reports No. 34/2012.
Abstract: Using techniques related to the notions of epigraph distance and Attouch-Wets-convergence, we show that under appropriate boundary conditions discrete elastica (i.e., polygonal curves of some fixed length that minimize a certain discrete bending energy) converge to smooth elastica (i.e., smooth curves of some given length that minimize smooth bending energy).

Variational Convergence and Discrete Minimal Surfaces
Henrik Schumacher.
Abstract: This work is concerned with the convergence behavior of the solutions to parametric variational problems. An emphasis is put on sequences of variational problems that arise as discretizations of either infinite-dimensional optimization problems or infinite-dimensional operator problems. Finally, the results are applied to discretizations of the Douglas-Plateau problem and of a boundary value problem in nonlinear elasticity.

  Seitenanfang  Impress 2017-03-31, Henrik Schumacher