We study the optimal rate of convergence in periodic homogenization of the viscous Hamilton-Jacobi equation $u^\varepsilon_t + H(\frac{x}{\varepsilon},Du^\varepsilon) = \varepsilon \Delta u^\varepsilon$ in $\mathbb R^n\times (0,\infty)$ subject to a given initial datum. We prove that $\|u^\varepsilon-u\|_{L^\infty(\mathbb R^n \times [0,T])} \leq C(1+T) \sqrt{\varepsilon}$ for any given $T>0$, where $u$ is the viscosity solution of the effective problem. Moreover, we show that the $O(\sqrt{\varepsilon})$ 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.

翻译：我们研究粘性Hamilton-Jacobi方程$u^\varepsilon_t + H(\frac{x}{\varepsilon},Du^\varepsilon) = \varepsilon \Delta u^\varepsilon$在$\mathbb R^n\times (0,\infty)$中给定初始数据条件下的周期均匀化最优收敛速度。对于任意给定的$T>0$，我们证明$\|u^\varepsilon-u\|_{L^\infty(\mathbb R^n \times [0,T])} \leq C(1+T) \sqrt{\varepsilon}$，其中$u$是有效问题的粘性解。此外，我们通过理论和数值实验表明，对于一类自然的$H$和Lipschitz连续的初始数据，$O(\sqrt{\varepsilon})$的收敛速度是最优的。当$H$一致凸时，能否提高收敛速度仍是一个有趣的问题。最后，我们基于近似校正问题的有限元逼近，提出了一种有效Hamiltonian的数值逼近格式。