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 $\Omega$ and a time $T$, the estimate gives an upper bound for the $L^\infty$-norm in $\Omega$ 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 $\Omega$. 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.