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

top
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

top

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

top
  1. F. Alouges and M. Pierre, Mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere. Numer. Math. To appear.  Zbl1088.65106
  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
  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.  Zbl1006.35050
  4. H. Brezis and J.-M. Coron, Large solutions for harmonic maps in two dimensions. Comm. Math. Phys.92 (1983) 203–215.  Zbl0532.58006
  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.  Zbl0911.65087
  6. K.-C. Chang, Heat flow and boundary value problem for harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire6 (1989) 363–395.  Zbl0687.58004
  7. J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds. Amer. J. Math.86 (1964) 109–160.  Zbl0122.40102
  8. A. Freire, Uniqueness for the harmonic map flow from surfaces to general targets. Comment. Math. Helv.70 (1995) 310–338.  Zbl0831.58018
  9. A. Freire, Uniqueness for the harmonic map flow in two dimensions. Calc. Var. Partial Differential Equations3 (1995) 95–105.  Zbl0814.35057
  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.  Zbl0913.65059
  11. M. Pierre, Weak BV convergence of moving finite elements for singular axisymmetric harmonic maps. SIAM J. Numer. Anal. To appear.  Zbl1109.65103
  12. E. Polak, Algorithms and consistent approximations, Optimization, Applied Mathematical Sciences124 (1997), Springer-Verlag, New York.  Zbl0899.90148
  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
  14. S. Rippa and B. Schiff, Minimum energy triangulations for elliptic problems. Comput. Methods Appl. Mech. Engrg.84 (1990) 257–274.  Zbl0742.65083
  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
  16. P. Topping, Reverse bubbling and nonuniqueness in the harmonic map flow. Internat. Math. Res. Notices10 (2002) 505–520.  Zbl1003.58014

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.