ALOUGES, Francois - Analyse Numérique et E.D.P., Université Paris-Sud, Bât. 425, 91405 Orsay cedex
PIERRE, Morgan - Analyse Numérique et E.D.P., Université Paris-Sud, Bât. 425, 91405 Orsay cedex

Abstract :

We propose a new viewpoint for the numerical computation of singular solutions to PDEs. The idea is to mesh the graph of the singular solution. We apply this idea to the test case of minimizing axi-symmetric harmonic maps from $B^2$ into $S^2$. The formulation of the PDE in terms of a minimization problem gives the criterion for the o ptimal mesh of the graph. In the first part we study the well-posedness of the continuous problem in the space of fun ctions with bounded energy. We introduce a lower semi-continuous extension of the energy w ith respect to the weak convergence in $BV$ and we prove that the extended minimization pr oblem admits a unique singular solution. In the second part we present the numerical results for various piecewise linear discretiza tions of the problem. The number of points in the mesh is fixed and the mesh is an unknown of the discrete minimization problem. This method allows us in particular to compute bound ary layers of zero thickness. Another paper, Part II, will prove the convergence of the method.

Article : Fichier Postscript
Contact : morgan.pierre@math.u-psud.fr