An Approximation Scheme for an Eikonal Equation with Discontinuous Coefficient

Abstract

We consider the stationary Hamilton--Jacobi equation $\sum_{i,j=1}^N b_{ij}(x)u_{x_i}u_{x_j}= \left[f(x)\right]^2$, in $\Omega$, where $\Omega$ is an open set of $\mathbb{R}^n$, $b$ 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 $L^1$. The last section contains some applications to control and image processing problems.