We compute numerically the minimizers of the Dirichlet energy
Keywords: harmonic maps, finite elements, mesh-refinement, Sobolev gradient, Newton algorithm, conjugate gradient
@article{COCV_2004__10_1_142_0, author = {Pierre, Morgan}, title = {Newton and conjugate gradient for harmonic maps from the disc into the sphere}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {142--167}, publisher = {EDP-Sciences}, volume = {10}, number = {1}, year = {2004}, doi = {10.1051/cocv:2003040}, mrnumber = {2084259}, zbl = {1076.65062}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2003040/} }
TY - JOUR AU - Pierre, Morgan TI - Newton and conjugate gradient for harmonic maps from the disc into the sphere JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2004 SP - 142 EP - 167 VL - 10 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2003040/ DO - 10.1051/cocv:2003040 LA - en ID - COCV_2004__10_1_142_0 ER -
%0 Journal Article %A Pierre, Morgan %T Newton and conjugate gradient for harmonic maps from the disc into the sphere %J ESAIM: Control, Optimisation and Calculus of Variations %D 2004 %P 142-167 %V 10 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2003040/ %R 10.1051/cocv:2003040 %G en %F COCV_2004__10_1_142_0
Pierre, Morgan. Newton and conjugate gradient for harmonic maps from the disc into the sphere. ESAIM: Control, Optimisation and Calculus of Variations, Volume 10 (2004) no. 1, pp. 142-167. doi : 10.1051/cocv:2003040. http://archive.numdam.org/articles/10.1051/cocv:2003040/
[1] A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708-1726. | Zbl
,[2] Numerical bifurcation of equilibria of nematic crystals between non-co-axial cylinders. Math. Models Methods Appl. Sci. 11 (2001) 459-473. | Zbl
and ,[3] Finite elements, in Theory, fast solvers, and applications in solid mechanics. Translated from the 1992 German edition by Larry L. Schumaker. Cambridge University Press, Cambridge, 2nd edn. (2001). | MR | Zbl
,[4] Analyse fonctionnelle. Masson (1996). | MR | Zbl
,[5] Large solutions for harmonic maps in two dimensions. Comm. Math. Phys. 92 (1983) 203-215. | Zbl
and ,[6] Finite-time blow up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36 (1992) 507-515. | Zbl
, and ,[7] Introduction à l'analyse numérique matricielle et à l'optimisation. Masson (1988). | Zbl
,[8] The physics of liquid crystals. Clarendon Press, Oxford (1993).
and ,[9] Function minimization by conjugate gradients. Comput. J. 7 (1994) 149-154. | Zbl
and ,[10] Cartesian currents in the calculus of variations. I. Springer-Verlag, Berlin (1998). | MR | Zbl
, and ,[11] Cartesian currents in the calculus of variations. II. Springer-Verlag, Berlin (1998). | MR | Zbl
, and ,[12] Singularities of harmonic maps. Bull. Amer. Math. Soc. (N.S.) 34 (1997) 15-34. | Zbl
,[13] Introduction à l'analyse non linéaire sur les variétés. Diderot Editeur Arts et Sciences (1987). | Zbl
,[14] Régularité des applications faiblement harmoniques entre une surface et une sphère. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 519-524. | Zbl
,[15] Symétries dans les problèmes variationnels et applications harmoniques. Istituti Editoriali e Poligrafici Internazionali, Pisa-Roma (1998).
,[16] Harmonic mappings betwenn surfaces. Springer-verlag, Lecture Notes in Math. 1062 (1984). | MR | Zbl
,[17] Riemannian Geometry. Walter de Gruyter (1995). | MR | Zbl
,[18] Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values. Manuscripta Math. 83 (1994) 31-38. | Zbl
,[19] Applications harmoniques de surfaces riemanniennes. J. Differ. Geom. 13 (1978) 51-78. | Zbl
,[20] Une méthode de gradient conjugué sur des variétés : application à certains problèmes de valeurs propres non linéaires. Numer. Funct. Anal. Optim. 1 (1979) 515-560. | Zbl
,[21] The concentration-compactness principle in the calculus of variations. The limit case, part 2. Rev. Mat. Iberoamericana 1 (1985) 45-121. | Zbl
,[22] Multiple integrals in the calculus of variations. Springer, New York (1966). | MR | Zbl
,[23] Sobolev gradients and boundary conditions for partial differential equations, in Recent developments in optimization theory and nonlinear analysis (Jerusalem, 1995), Amer. Math. Soc., Providence, RI. Contemp. Math. 204 (1997) 171-181 | Zbl
,[24] Optimization, Appl. Math. Sci. 124 (1997). | Zbl
,[25] Remark on the Dirichlet problem for harmonic maps from the disc into the 2-sphere. Proc. R. Soc. Edinb. 122A (1992) 63-67. | Zbl
,[26] Boundary regularity of weakly harmonic maps from surfaces. J. Funct. Anal. 114 (1993) 63-67. | Zbl
,[27] Boundary regularity and the Dirichlet problem for harmonic maps. J. Dif. Geom. 18 (1983) 253-268. | Zbl
and ,[28] Triangle: engineering a 2d quality mesh generator and delaunay triangulator. http://www-2.cs.cmu.edu/quake/triangle.html.
,[29] An introduction to the conjugate gradient method without the agonizing pain. http://www-2.cs.cmu.edu/jrs/jrspapers.html#cg (1994).
,[30] The Dirichlet problem for harmonic maps from the disc into the 2-sphere. Proc. R. Soc. Edinb. 113A (1989) 229-234. | Zbl
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