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 $T>0$. Moreover, we show that the $O(\sqrt{\varepsilon})$ rate is optimal in general, both theoretically and through numerical experiments. 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：本文研究粘性Hamilton-Jacobi方程$u^\varepsilon_t + H(\frac{x}{\varepsilon},Du^\varepsilon) = \varepsilon \Delta u^\varepsilon$在$\mathbb R^n\times (0,\infty)$上满足给定初始数据的周期均匀化问题的最优收敛速率。我们证明了对任意给定的$T>0$，有$\|u^\varepsilon-u\|_{L^\infty(\mathbb R^n \times [0,T])} \leq C(1+T) \sqrt{\varepsilon}$。此外，我们通过理论分析和数值实验表明$O(\sqrt{\varepsilon})$的速率在一般情况下是最优的。最后，我们提出一种基于逼近校正子问题有限元近似的有效Hamilton量数值计算方案。