An approximation scheme for a Hamilton-Jacobi equation defined on a network

In this paper we study approximation of Hamilton–Jacobi equations defined on a network. We introduce an appropriate notion of viscosity solution on networks which satisfies existence, uniqueness and stability properties. Then we define an approximation scheme of semi-Lagrangian type by discretizing in time the representation formula for the solution of Hamilton–Jacobi equations and we prove that the discrete problem admits a unique solution. Moreover we prove that the solution of the approximation scheme converges to the solution of the continuous problem uniformly on the network. In the second part of the paper we study a fully discrete scheme obtained via a finite elements discretization of the semi-discrete problem. Also for fully discrete scheme we prove the well posedness and the convergence to the viscosity solution of the Hamilton–Jacobi equation. We also discuss some issues concerning the implementation of the algorithm and we present some numerical examples.