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

TL;DR

This paper establishes the optimal convergence rate of order er or viscous Hamilton-Jacobi equations in periodic homogenization, demonstrating both theoretical optimality and numerical methods for approximation.

Contribution

It proves the er or the convergence rate in periodic homogenization of viscous Hamilton-Jacobi equations and introduces a numerical scheme for approximating the effective Hamiltonian.

Findings

Convergence rate of er or solutions is er or Lipschitz initial data.

The er or rate is shown to be optimal.

Numerical experiments support the theoretical results.

Abstract

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.