Local a posteriori error estimates for time-dependent Hamilton-Jacobi equations

In this paper, we obtain the first local a posteriori error estimate for time-dependent Hamilton-Jacobi equations. Given an arbitrary domain Ω and a time T, the estimate gives an upper bound for the L^∞-norm in Ω at time T of the difference between the viscosity solution u and any continuous function v in terms of the initial error in the domain of dependence and in terms of the (shifted) residual of v in the union of all the cones of dependence with vertices in Ω. The estimate holds for general Hamiltonians and any space dimension. It is thus an ideal tool for devising adaptive algorithms with rigorous error control for time-dependent Hamilton-Jacobi equations. This result is an extension to the global a posteriori error estimate obtained by S. Albert, B. Cockburn, D. French, and T. Peterson in A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations. Part II: The time-dependent case, Finite Volumes for Complex Applications, vol. III, June 2002, pp. 17-24. Numerical experiments investigating the sharpness of the a posteriori error estimates are given.