Optimal Rate of Convergence in Periodic Homogenization of Viscous Hamilton-Jacobi Equations
Jianliang Qian, Timo Sprekeler, Hung V. Tran, Yifeng Yu · Multiscale Modeling and Simulation · 2024
Abstract. We study the optimal rate of convergence in periodic homogenization of the viscous Hamilton–Jacobi equation [Formula: see text] in [Formula: see text] subject to a given initial datum. We prove that [Formula: see text] for any given [Formula: see text], where [Formula: see text] is the viscosity solution of the effective problem. Moreover, we show that the [Formula: see text] rate is optimal for a natural class of [Formula: see text] 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 [Formula: see text] 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.