An approximation scheme for an Hamilton-Jacobi equation defined on a   network
  In this paper we study an approximation scheme for an Hamilton-Jacobi equation of Eikonal type defined on a network. We introduce an appropriate notion of viscosity solution for this class of equations (see \cite{sc}) and we prove that an approximation scheme of semi-Lagrangian type converges to the unique solution of the problem. 
Fabio Camilli, Adriano Festa and Dirk Schieborn
math.AP math.NA