A convergent scheme for a non-local coupled system modelling dislocations densities dynamics

Ahmad El Hajj (1) , Nicolas Forcadel (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Within the framework of viscosity solutions, we prove a long time existence and uniqueness result for the solution of this model. We also propose a convergent numerical scheme and we prove a Crandall-Lions type error estimate between the continuous solution and the numerical one. As far as we know, this is the first error estimate of Crandall-Lions type for Hamilton-Jacobi systems. We also provide some numerical simulations.

Mots clés: system; Hamilton Jacobi equations; viscosity solutions; dislocations densities dynamics; numerical scheme; error estimate; system.

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

Ahmad El Hajj, Nicolas Forcadel. A convergent scheme for a non-local coupled system modelling dislocations densities dynamics. Mathematics of Computation, 2008, 77, pp.789--812. ⟨hal-00415944⟩