Raisch, Alexander Helmut-Wilhelm: Finite Element Methods for Geometric Problems. - Bonn, 2012. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-30200
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-30200,
author = {{Alexander Helmut-Wilhelm Raisch}},
title = {Finite Element Methods for Geometric Problems},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2012,
month = oct,

note = {In the herewith presented work we numerically treat geometric partial differential equations using finite element methods. Problems of this type appear in many applications from physics, biology and engineering use.
We may partition the work in two blocks. The first one, including the chapters two to five, is about the approximation of stationary points of conformally invariant, nonlinear, elliptic energy functionals. Main interest is a compactness result for accumulation points of their discrete counterparts. The corresponding Euler-Lagrange equations are nonlinear, elliptic and of second order. They contain critical nonlinearities that are quadratic in the first derivatives. Thus, accumulation points of solutions to the discrete problem are not solutions of the continuous problem in general. We deduce a weak formulation in a mixed form and chose appropriate spaces for the discretization. First we show existence of discrete solutions and then, by the use of compensated compactness and standard finite element arguments, we establish convergence. Finally we introduce an iterative algorithm for the numerical realization and run different simulations. Hereby we confirm theoretical predictions derived in the stability analysis.
The second part is about the derivation of gradient flows for shape functionals and their discretization with parametric finite elements. First, we consider the Willmore energy of a twodimensional surface in the threedimensional ambient space and deduce its first variation. Afterwards we phrase the corresponding gradient flow in a weak form and discuss possible discretizations. During the further progress of the work we modell cell membranes and the effects of surface active agents on the shape of these cells. Numerical simulations with closed surface give promising results and a reason to intensify the research in this field.},

url = {http://hdl.handle.net/20.500.11811/5398}

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