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Newton and conjugate gradient for harmonic mapsfrom the disc into the sphere

Published online by Cambridge University Press:  15 February 2004

Morgan Pierre
Morgan Pierre*
Affiliation:
Centre de Mathématiques et de Leurs Applications, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France; [email protected].
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Abstract

We compute numerically the minimizers of the Dirichlet energy $$E(u)=\frac{1}{2}\int_{B^2}|\nabla u|^2 {\rm d}x$$ among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P 1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned projected gradient). In order to improve the convergence, we generalize to manifolds the classical Newton and conjugate gradient algorithms. We give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting.

Keywords

Harmonic mapsfinite elementsmesh-refinementSobolev gradientNewton algorithmconjugate gradient.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 1 , January 2004 , pp. 142 - 167
Copyright
© EDP Sciences, SMAI, 2004

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