A posteriori error estimates for time-dependent Hamilton-Jacobi equations.

Abstract

We present a local a posteriori error estimate for general numerical methods for time-dependent Hamilton-Jacobi equations. Since Hamilton-Jacobi equations find applications in many areas there is an interest in constructing efficient algorithms that produce numerical approximations with a guaranteed precision set beforehand by the practitioner. To develop such algorithms, it is important to be able to estimate the quality of any given approximation in terms of computable quantities only, and this is what a posteriori error estimates provide. Given an arbitrary domain 2 Rd and a time T > 0, the a posteriori error estimate gives an upper bound for the L1-norm of the difference between the viscosity solution u and any continuous function v in at time T. The estimate holds for general Hamiltonians and any space dimensions d. The case = Rd reduces 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 both the global and local a posteriori error estimates are provided. The results confirm that the a posteriori error estimates are very efficient and are thus an ideal tool for devising adaptive algorithms with rigorous error control for time-dependent Hamilton-Jacobi equations.