Mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere

We describe in a mathematical setting the singular energy minimizing axisymmetric harmonic maps from the unit disc into the unit sphere; then, we use this as a test case to compute optimal meshes in presence of sharp boundary layers. For the well-posedness of the continuous minimizing problem, we introduce a lower semicontinuous extension of the energy with respect to weak convergence in BV, and we prove that the extended minimization problem has a unique singular solution. We then show how a moving finite element method, in which the mesh is an unknown of the discrete minimization problem obtained by finite element discretization, mimics this geometric point of view. Finally, we present numerical computations with boundary layers of zero thickness, and we give numerical evidence of the convergence of the method. This last aspect is proved in another paper. This work was supported by the Centre de Mathématiques et de Leurs Applications, Ecole Normale Supérieure de Cachan, 61 av. du Président Wilson, 94235 Cachan Cedex, France