Dislocation dynamics were studied from a level-set point-of-view. The model presented considered the zero level-set of the solution of a non-local Hamilton Jacobi equation to be a dislocation on a plane of a crystal. The front had a normal speed, depending upon the solution itself. Existence and uniqueness for short times in the set of continuous viscosity solutions was proved. A first-order finite difference scheme for the corresponding level set formulation of the model was also presented. The scheme was based upon the Osher-Sethian monotone numerical Hamiltonian. The non-local character of the problem made it non-monotone. An explicit convergence rate of the approximate solution to the viscosity solution was obtained.

A Convergent Scheme for a Non-Local Hamilton Jacobi Equation Modelling Dislocation Dynamics. O.Alvarez, E.Carlini, R.Monneau, E.Rouy: Numerische Mathematik, 2006, 104[4], 413-44