Moving mesh for the axisymmetric harmonic map flow

Benoit Merlet; Morgan Pierre

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 L2-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 39.4 (2010): 781-796. <http://eudml.org/doc/194286>.

@article{Merlet2010,
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 L2-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},
keywords = {Moving mesh; finite elements; harmonic map flow; axisymmetric.; moving mesh; axisymmetric; corotational initial data; corotational degree},
language = {eng},
month = {3},
number = {4},
pages = {781-796},
publisher = {EDP Sciences},
title = {Moving mesh for the axisymmetric harmonic map flow},
url = {http://eudml.org/doc/194286},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Merlet, Benoit
AU - Pierre, Morgan
TI - Moving mesh for the axisymmetric harmonic map flow
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
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 L2-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.; moving mesh; axisymmetric; corotational initial data; corotational degree
UR - http://eudml.org/doc/194286
ER -

References

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  10. F. Hülsemann and Y. Tourigny, A new moving mesh algorithm for the finite element solution of variational problems. SIAM J. Numer. Anal.35 (1998) 1416–1438.  
  11. M. Pierre, Weak BV convergence of moving finite elements for singular axisymmetric harmonic maps. SIAM J. Numer. Anal. To appear.  
  12. E. Polak, Algorithms and consistent approximations, Optimization, Applied Mathematical Sciences124 (1997), Springer-Verlag, New York.  
  13. J. Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom.3 (1995) 297–315.  
  14. S. Rippa and B. Schiff, Minimum energy triangulations for elliptic problems. Comput. Methods Appl. Mech. Engrg.84 (1990) 257–274.  
  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.  
  16. P. Topping, Reverse bubbling and nonuniqueness in the harmonic map flow. Internat. Math. Res. Notices10 (2002) 505–520.  

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