# 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 P1 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 (2010): 142-167. <http://eudml.org/doc/90717>.

@article{Pierre2010,
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 P1 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.},
language = {eng},
month = {3},
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/90717},
volume = {10},
year = {2010},
}

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
DA - 2010/3//
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 $u:B^2\to S^2$ 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 P1 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.
UR - http://eudml.org/doc/90717
ER -

## References

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