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

Pierre, Morgan

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 1, pp. 142-167.

We compute numerically the minimizers of the Dirichlet energy

$$E\left(u\right)=\frac{1}{2}{\int }_{B^2}{\left|\nabla u\right|}^2\mathrm{d}x$$

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.

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