A Generalized Fast Marching Method for dislocation dynamics

Elisabetta Carlini (1) , Nicolas Forcadel (2) , Régis Monneau (3)

1 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

In this paper, we consider a Generalized Fast Marching Method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hyper-surface in $\mathbb R^N$ (with $N=2$ for physical applications) is given by its normal velocity which is a non-local function of the whole shape of the hyper-surface itself. For this dynamics, we show a convergence result of the GFMM as the mesh size goes to zero. We also provide some numerical simulations in dimension $N=2$.

Mots clés: non-local equations; Hamilton-Jacobi equations; fast marching scheme; convergence; viscosity solutions; dislocation dynamics; non-local equations.

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Elisabetta Carlini, Nicolas Forcadel, Régis Monneau. A Generalized Fast Marching Method for dislocation dynamics. SIAM Journal on Numerical Analysis, 2011, 49 (6), pp.2470-2500. ⟨10.1137/090770862⟩. ⟨hal-00415902v2⟩