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Moving mesh for the axisymmetric harmonic map flow

Published online by Cambridge University Press:  15 August 2005

Benoit Merlet  and
Morgan Pierre
Benoit Merlet
Affiliation:
ENS Cachan, Antenne de Bretagne, Avenue Robert Schumann, 35170 Bruz France. benoit.merlet@bretagne.ens-cachan.fr
Morgan Pierre
Affiliation:
Laboratoire de Mathématiques, Université de Poitiers, Bd Marie et Pierre Curie, BP 30179, 86962 Futuroscope Cedex, France. Morgan.Pierre@math.univ-poitiers.fr Centre de Mathématiques et de Leurs Applications, École Normale Supérieure de Cachan, 61 Av. du Président Wilson, 94235 Cachan Cedex, France.
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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.

Keywords

Moving meshfinite elementsharmonic map flowaxisymmetric.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 4 , July 2005 , pp. 781 - 796
Copyright
© EDP Sciences, SMAI, 2005

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References

F. Alouges and M. Pierre, Mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere. Numer. Math. To appear.
Bethuel, F., Coron, J.-M., Ghidaglia, J.-M. and Soyeur, A., 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.
Bertsch, M., Dal Passo, R. and van der Hout, R., Nonuniqueness for the heat flow of harmonic maps on the disk. Arch. Rational Mech. Anal. 161 (2002) 93–112.
Brezis, H. and Coron, J.-M., Large solutions for harmonic maps in two dimensions. Comm. Math. Phys. 92 (1983) 203–215. CrossRef
Carlson, N. and Miller, K., Design and application of a gradient-weighted moving finite element code. I. In one dimension. SIAM J. Sci. Comput. 19 (1998) 728–765. CrossRef
Chang, K.-C., Heat flow and boundary value problem for harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 363–395. CrossRef
Eells, J. and Sampson, J., Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964) 109–160. CrossRef
Freire, A., Uniqueness for the harmonic map flow from surfaces to general targets. Comment. Math. Helv. 70 (1995) 310–338. CrossRef
Freire, A., Uniqueness for the harmonic map flow in two dimensions. Calc. Var. Partial Differential Equations 3 (1995) 95–105. CrossRef
Hülsemann, F. and Tourigny, Y., A new moving mesh algorithm for the finite element solution of variational problems. SIAM J. Numer. Anal. 35 (1998) 1416–1438.
M. Pierre, Weak BV convergence of moving finite elements for singular axisymmetric harmonic maps. SIAM J. Numer. Anal. To appear.
E. Polak, Algorithms and consistent approximations, Optimization, Applied Mathematical Sciences 124 (1997), Springer-Verlag, New York.
Qing, J., On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3 (1995) 297–315. CrossRef
Rippa, S. and Schiff, B., Minimum energy triangulations for elliptic problems. Comput. Methods Appl. Mech. Engrg. 84 (1990) 257–274. CrossRef
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.
Topping, P., Reverse bubbling and nonuniqueness in the harmonic map flow. Internat. Math. Res. Notices 10 (2002) 505–520. CrossRef