An approximation scheme for an eikonal equation with discontinuous coefficient

We consider the stationary Hamilton–Jacobi equation N i,j=1 bij (x)uxiuxj = [f(x)]2, in Ω, where Ω is an open set of Rn, b can vanish at some points, and the right-hand-side f is strictly positive and is allowed to be discontinuous. More precisely, we consider a special class of discontinuities for which the notion of viscosity solution is well-suited. We propose a semi-Lagrangian scheme for the numerical approximation of the viscosity solution in the sense of Ishii and we study its properties. We also prove an a priori error estimate for the scheme in L1. The last section contains some applications to control and image processing problems.