Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities

Abstract

We consider the Hamilton-Jacobi equation of eikonal type H(∇u) = f(x), x ∈ Ω, where H is convex and f is allowed to be discontinuous. Under a suitable assumption on f we prove a comparison principle for viscosity sub- and supersolutions in the sense of Ishii. Furthermore, we develop an error analysis for a class of finite difference schemes, which are monotone, consistent and satisfy a suitable stability condition.