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.

We compute numerically the minimizers of the Dirichlet energy

E(u)=1 2 B 2 |u| 2 dx
among maps u:B 2 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.

DOI: 10.1051/cocv:2003040
Classification: 58E20, 78M10, 65N30, 90C53
Keywords: harmonic maps, finite elements, mesh-refinement, Sobolev gradient, Newton algorithm, conjugate gradient
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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, Volume 10 (2004) no. 1, pp. 142-167. doi : 10.1051/cocv:2003040. http://archive.numdam.org/articles/10.1051/cocv:2003040/

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