Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations

We study the optimal rate of convergence in periodic homogenization of the viscous Hamilton-Jacobi equation $u_t^{\epsilon }+H(\frac{x}{\epsilon },D{u}^{\epsilon })=\epsilon \mathrm{\Delta }{u}^{\epsilon }$ in ${\mathbb{R}}^n\times (0,\mathrm{\infty })$ subject to a given initial datum. We prove that $\mathrm{\parallel }{u}^{\epsilon }-u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n\times [0,T])}\le C(1+T)\sqrt{\epsilon }$ for any given $T>0$, where $u$ is the viscosity solution of the effective problem. Moreover, we show that the $O(\sqrt{\epsilon })$ rate is optimal for a natural class of $H$ and a Lipschitz continuous initial datum, both theoretically and through numerical experiments. It remains an interesting question to investigate whether the convergence rate can be improved when $H$ is uniformly convex. Finally, we propose a numerical scheme for the approximation of the effective Hamiltonian based on a finite element approximation of approximate corrector problems.