# Newton and conjugate gradient for harmonic maps from the disc into the sphere

• Volume: 10, Issue: 1, page 142-167
• ISSN: 1292-8119

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## Abstract

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We compute numerically the minimizers of the Dirichlet energy$E\left(u\right)=\frac{1}{2}{\int }_{{B}^{2}}{|\nabla u|}^{2}\mathrm{d}x$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.

## How to cite

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Pierre, Morgan. "Newton and conjugate gradient for harmonic maps from the disc into the sphere." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2004): 142-167. <http://eudml.org/doc/245455>.

@article{Pierre2004,
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 $u:B^2\rightarrow S^2$ 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.},
author = {Pierre, Morgan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {harmonic maps; finite elements; mesh-refinement; Sobolev gradient; Newton algorithm; conjugate gradient; Harmonic maps; preconditioning; convergence},
language = {eng},
number = {1},
pages = {142-167},
publisher = {EDP-Sciences},
title = {Newton and conjugate gradient for harmonic maps from the disc into the sphere},
url = {http://eudml.org/doc/245455},
volume = {10},
year = {2004},
}

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
PB - EDP-Sciences
VL - 10
IS - 1
SP - 142
EP - 167
AB - 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 $u:B^2\rightarrow S^2$ 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.
LA - eng
KW - harmonic maps; finite elements; mesh-refinement; Sobolev gradient; Newton algorithm; conjugate gradient; Harmonic maps; preconditioning; convergence
UR - http://eudml.org/doc/245455
ER -

## References

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