Finite difference scheme for the Willmore flow of graphs

Tomáš Oberhuber

Kybernetika (2007)

  • Volume: 43, Issue: 6, page 855-867
  • ISSN: 0023-5954

Abstract

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In this article we discuss numerical scheme for the approximation of the Willmore flow of graphs. The scheme is based on the finite difference method. We improve the scheme we presented in Oberhuber [Obe-2005-2,Obe-2005-1] which is based on combination of the forward and the backward finite differences. The new scheme approximates the Willmore flow by the central differences and as a result it better preserves symmetry of the solution. Since it requires higher regularity of the solution, additional numerical viscosity is necessary in some cases. We also present theorem showing stability of the scheme together with the EOC and several results of the numerical experiments.

How to cite

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Oberhuber, Tomáš. "Finite difference scheme for the Willmore flow of graphs." Kybernetika 43.6 (2007): 855-867. <http://eudml.org/doc/33902>.

@article{Oberhuber2007,
abstract = {In this article we discuss numerical scheme for the approximation of the Willmore flow of graphs. The scheme is based on the finite difference method. We improve the scheme we presented in Oberhuber [Obe-2005-2,Obe-2005-1] which is based on combination of the forward and the backward finite differences. The new scheme approximates the Willmore flow by the central differences and as a result it better preserves symmetry of the solution. Since it requires higher regularity of the solution, additional numerical viscosity is necessary in some cases. We also present theorem showing stability of the scheme together with the EOC and several results of the numerical experiments.},
author = {Oberhuber, Tomáš},
journal = {Kybernetika},
keywords = {Willmore flow; method of lines; curvature minimization; gradient flow; Laplace–Beltrami operator; Gauss curvature; central differences; numerical viscosity; Willmore flow; method of lines; curvature minimization; gradient flow; Laplace-Beltrami operator; Gauss curvature; central differences; numerical viscosity},
language = {eng},
number = {6},
pages = {855-867},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Finite difference scheme for the Willmore flow of graphs},
url = {http://eudml.org/doc/33902},
volume = {43},
year = {2007},
}

TY - JOUR
AU - Oberhuber, Tomáš
TI - Finite difference scheme for the Willmore flow of graphs
JO - Kybernetika
PY - 2007
PB - Institute of Information Theory and Automation AS CR
VL - 43
IS - 6
SP - 855
EP - 867
AB - In this article we discuss numerical scheme for the approximation of the Willmore flow of graphs. The scheme is based on the finite difference method. We improve the scheme we presented in Oberhuber [Obe-2005-2,Obe-2005-1] which is based on combination of the forward and the backward finite differences. The new scheme approximates the Willmore flow by the central differences and as a result it better preserves symmetry of the solution. Since it requires higher regularity of the solution, additional numerical viscosity is necessary in some cases. We also present theorem showing stability of the scheme together with the EOC and several results of the numerical experiments.
LA - eng
KW - Willmore flow; method of lines; curvature minimization; gradient flow; Laplace–Beltrami operator; Gauss curvature; central differences; numerical viscosity; Willmore flow; method of lines; curvature minimization; gradient flow; Laplace-Beltrami operator; Gauss curvature; central differences; numerical viscosity
UR - http://eudml.org/doc/33902
ER -

References

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  5. Du Q., Liu, C., Wang X., A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, J. Comput. Phys. (2004), 198, 450–468 (198,) MR2062909
  6. Dziuk G., Kuwert, E., Schätzle R., Evolution of elastic curves in ℝ n : Existence and Computation, SIAM J. Math. Anal. 41 (2003), 6, 2161–2179 MR2034610
  7. Kuwert E., Schätzle R., The Willmore flow with small initial energy, J. Differential Geom. 57 (2001), 409–441 Zbl1035.53092MR1882663
  8. Minárik V., Kratochvíl, J., Mikula K., Numerical simulation of dislocation dynamics by means of parametric approach, In: Proc. Czech Japanese Seminar in Applied Mathematics (M. Beneš, J. Mikyška, and T. Oberhuber, eds.), Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague 2005, pp. 128–138 
  9. Oberhuber T., Computational study of the Willmore flow on graphs, Accepted to the Proc. Equadiff 11, 2005 
  10. Oberhuber T., Numerical solution for the Willmore flow of graphs, In: Proc. Czech–Japanese Seminar in Applied Mathematics 2005 (M. Beneš, M. Kimura and T. Nakaki, eds.), COE Lecture Note Vol. 3, Faculty of Mathematics, Kyushu University Fukuoka, October 2006, ISSN 1881-4042, pp. 126–138 Zbl1145.65323MR2279053
  11. Simonett G., The Willmore flow near spheres, Differential and Integral Equations 14 (2001), No. 8, 1005–1014 Zbl1161.35429MR1827100
  12. Willmore T. J., Riemannian Geometry, Oxford University Press, Oxford 1997 Zbl1117.53017MR1261641

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