WS 2010/11
Analysis und Numerik geometrischer Differentialgleichungen
Klaus Ecker und Ralf Kornhuber
Termine:
Dienstags, 12:15 - 13:45 Uhr, Arnimallee 6, Raum 007/008
Erster Vortrag: 19. Oktober 2010
Vorträge
19.10.2010
K. Ecker: Geometrical analysis of mean curvature flow.
26.10.2010
R. Kornhuber: Numerical solution of elliptic and parabolic problems.
1.5 + 4.6
26.10.2010
Von Deylen: Computational mean curvature flow: Parametric approach. [2], Sec. 4. and 5.
Here is an C++ example program for the presented semi-implicit parametric scheme. It evolves a given graph surface and prints its output as 2D plot with color-coded height or as PLY or MatLab files (meaning that visualization is up to you, consider e. g. MeshViewer for the PLY files). The code should run with VC++ and g++ (makefile included), further documentation is in the main cpp file. The code stems mostly from a programming lab at Bonn University, therefore an explicit gradient descent algorithm is also included. If you have any questions,do not hesitate to ask; I will be glad to help you (contact).
02.11.2010
Wolf: Computational mean curvature flow: Implicit surfaces. [2], Sec. 6 and [3].
09.11.2010
C. Gräser: Computational mean curvature flow: Phase field approach. [2], Sec. 7.
16.11.2010
N.N.: Computational inverse mean curvature flow. [8].
23.11.2010
Jachan/Hardering/Smith: Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature. [4].
30.11.2010
Jachan/Hardering: Numerical computation of constant mean curvature surfaces I. [5].
07.12.2010
Jachan/Hardering: Numerical computation of constant mean curvature surfaces II. [5].
14.12.2010
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04.01.2011
Tobias Marxen: Ricci flow on warped product manifolds.
11.01.2011
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18.01.2011
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25.01.2011
S.W. von Deylen: The metric distortion tensor equation and convergence of weak curvatures
8.02.2011
G. Wheeler: Fourth order geometric flows
15.02.2011
A. Afuni: Finite elements on evolving surfaces
Literatur:
- [1]
- Gerd Dziuk, Finite Elements for the Beltrami operator on arbitrary surfaces. In: S. Hildebrandt, R. Leis (eds.) Partial Differential Equations and Calculus of Variations, (1988), 483-490.
- [2]
- Klaus Deckelnick, Gerd Dziuk, Charles M. Elliott, Computation of geometric partial differential equations and mean curvature flow. Acta Numerica (2005), 139-232.
- [3]
- Klaus Deckelnick, Gerd Dziuk Charles M. Elliott, C.-J. Heine, An h-narrow band finite-element method for elliptic equations on implicit surfaces. IMA J. Numer. Anal. (2010) 30, 351-376.
- [4]
- Jan Metzger, Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature, J. Differential Geom. Volume 77, Number 2 (2007), 201-236.
- [5]
- Jan Metzger, Numerical computation of constant mean curvature surfaces using finite elements, Classical and Quantum Gravity, Volume 21, Number 19.
- [6]
- Lars Anderson, Jan Metzger, The Area of Horizons and the Trapped Region, Comm. Math. Phys. Vol. 290, 2009, 941-972.
- [7]
- Bernhard Hein, A homotopy approach to solving the inverse mean curvature flow, Calc. Var, Vol. 28 No. 2, 2007, 249-273.
- [8]
- Pasch, E.: Numerische Verfahren zur Berechnung von Krümmungsflüssen. PhD Thesis, Universität Tübingen, October 1998.
- [9]
- John W. Barrett, Harald Garcke, Robert Nürnberg, On the parametric finite element approximation of evolving hypersurfaces in R3, Journal of Computational Physics archive, Volume 227, 2008, 4281-4307.