Moving mesh for the axisymmetric harmonic map flow

Benoit Merlet; Morgan Pierre

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 4, page 781-796
  • ISSN: 0764-583X

Abstract

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We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L 2 -gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method.

How to cite

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Merlet, Benoit, and Pierre, Morgan. "Moving mesh for the axisymmetric harmonic map flow." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.4 (2005): 781-796. <http://eudml.org/doc/245389>.

@article{Merlet2005,
abstract = {We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the $\mathrm \{L\}^\{2\}$-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method.},
author = {Merlet, Benoit, Pierre, Morgan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {moving mesh; finite elements; harmonic map flow; axisymmetric; corotational initial data; corotational degree},
language = {eng},
number = {4},
pages = {781-796},
publisher = {EDP-Sciences},
title = {Moving mesh for the axisymmetric harmonic map flow},
url = {http://eudml.org/doc/245389},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Merlet, Benoit
AU - Pierre, Morgan
TI - Moving mesh for the axisymmetric harmonic map flow
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 4
SP - 781
EP - 796
AB - We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the $\mathrm {L}^{2}$-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method.
LA - eng
KW - moving mesh; finite elements; harmonic map flow; axisymmetric; corotational initial data; corotational degree
UR - http://eudml.org/doc/245389
ER -

References

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  2. [2] F. Bethuel, J.-M. Coron, J.-M. Ghidaglia and A. Soyeur, Heat flows and relaxed energies for harmonic maps, in Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), Birkhäuser Boston, Boston, MA. Progr. Nonlinear Differential Equations Appl. 7 (1992) 99–109. Zbl0795.35053
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  11. [11] M. Pierre, Weak BV convergence of moving finite elements for singular axisymmetric harmonic maps. SIAM J. Numer. Anal. To appear. Zbl1109.65103MR2182135
  12. [12] E. Polak, Algorithms and consistent approximations, Optimization, Applied Mathematical Sciences 124 (1997), Springer-Verlag, New York. Zbl0899.90148MR1454128
  13. [13] J. Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3 (1995) 297–315. Zbl0868.58021
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  15. [15] M. Struwe, The evolution of harmonic maps, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990). Math. Soc. Japan (1991) 1197–1203. Zbl0744.58011
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