Geodesic
Moving mesh for the axisymmetric harmonic map flow
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 39 (2005) no. 4, pp. 781-796.

Voir la notice de l'article provenant de la source Numdam

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.

DOI : 10.1051/m2an:2005034
Classification : 35A05, 35K55, 65N30, 65N50, 65N99
Mots-clés : moving mesh, finite elements, harmonic map flow, axisymmetric 
@article{M2AN_2005__39_4_781_0,
     author = {Merlet, Benoit and Pierre, Morgan},
     title = {Moving mesh for the axisymmetric harmonic map flow},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {781--796},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {4},
     year = {2005},
     doi = {10.1051/m2an:2005034},
     mrnumber = {2165679},
     zbl = {1078.35008},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2005034/}
}
TY  - JOUR
AU  - Merlet, Benoit
AU  - Pierre, Morgan
TI  - Moving mesh for the axisymmetric harmonic map flow
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2005
SP  - 781
EP  - 796
VL  - 39
IS  - 4
PB  - EDP-Sciences
UR  - https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2005034/
DO  - 10.1051/m2an:2005034
LA  - en
ID  - M2AN_2005__39_4_781_0
ER  - 
%0 Journal Article
%A Merlet, Benoit
%A Pierre, Morgan
%T Moving mesh for the axisymmetric harmonic map flow
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2005
%P 781-796
%V 39
%N 4
%I EDP-Sciences
%U https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2005034/
%R 10.1051/m2an:2005034
%G en
%F M2AN_2005__39_4_781_0
Merlet, Benoit; Pierre, Morgan. Moving mesh for the axisymmetric harmonic map flow. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 39 (2005) no. 4, pp. 781-796. doi : 10.1051/m2an:2005034. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2005034/

[1] F. Alouges and M. Pierre, Mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere. Numer. Math. To appear. | Zbl | MR

[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. | Zbl

[3] M. Bertsch, R. Dal Passo and R. Van Der Hout, Nonuniqueness for the heat flow of harmonic maps on the disk. Arch. Rational Mech. Anal. 161 (2002) 93-112. | Zbl

[4] H. Brezis and J.-M. Coron, Large solutions for harmonic maps in two dimensions. Comm. Math. Phys. 92 (1983) 203-215. | Zbl

[5] N. Carlson and K. Miller, Design and application of a gradient-weighted moving finite element code. I. In one dimension. SIAM J. Sci. Comput. 19 (1998) 728-765. | Zbl

[6] K.-C. Chang, Heat flow and boundary value problem for harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 363-395. | mathdoc-id | Zbl | EuDML

[7] J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964) 109-160. | Zbl

[8] A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets. Comment. Math. Helv. 70 (1995) 310-338. | Zbl | EuDML

[9] A. Freire, Uniqueness for the harmonic map flow in two dimensions. Calc. Var. Partial Differential Equations 3 (1995) 95-105. | Zbl

[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. | Zbl

[11] M. Pierre, Weak BV convergence of moving finite elements for singular axisymmetric harmonic maps. SIAM J. Numer. Anal. To appear. | Zbl | MR

[12] E. Polak, Algorithms and consistent approximations, Optimization, Applied Mathematical Sciences 124 (1997), Springer-Verlag, New York. | Zbl | MR

[13] J. Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3 (1995) 297-315. | Zbl

[14] S. Rippa and B. Schiff, Minimum energy triangulations for elliptic problems. Comput. Methods Appl. Mech. Engrg. 84 (1990) 257-274. | Zbl

[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. | Zbl

[16] P. Topping, Reverse bubbling and nonuniqueness in the harmonic map flow. Internat. Math. Res. Notices 10 (2002) 505-520. | Zbl

Cité par Sources :