A posteriori error estimates for the effective Hamiltonian of dislocation dynamics

Simone Cacace (1, 2) , Antonin Chambolle (1) , Régis Monneau (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

We study an implicit and discontinuous scheme for a non-local Hamilton-Jacobi equation modelling dislocation dynamics. For the evolution problem, we prove an a posteriori estimate of Crandall-Lions type for the error between continuous and discrete solutions. We deduce an a posteriori error estimate for the effective Hamiltonian associated to a stationary cell problem. In dimension one and under suitable assumptions, we also give improved a posteriori estimates. Numerical simulations are provided.

Mots clés: dislocation dynamics; Hamilton-Jacobi; implicit scheme; a posteriori error estimate; effective Hamiltonian; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Simone Cacace, Antonin Chambolle, Régis Monneau. A posteriori error estimates for the effective Hamiltonian of dislocation dynamics. Numerische Mathematik, 2011, 55 p. ⟨10.1007/s00211-011-0430-z⟩. ⟨hal-00432605⟩