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

Abstract

We study the optimal rate of convergence in periodic homogenization of the viscous Hamilton-Jacobi equation $u^\varepsilon_t + H(\frac{x}{\varepsilon},Du^\varepsilon) = \varepsilon Δu^\varepsilon$ 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.

Figures (4)

• Figure 5.1: Illustration of the error $\|u^{\varepsilon}_{\Delta x}(\cdot,1)-u(\cdot,1)\|_{L^\infty(\Omega)}$ for Examples \ref{['ex:1']}--\ref{['ex:5']}.
• Figure 5.2: Illustration of $\|u^{\varepsilon}_{\Delta x}(\cdot,T)-u^{\varepsilon/2}_{\Delta x}(\cdot,T)\|_{L^\infty(\Omega)}$ for Examples \ref{['ex:6']}--\ref{['ex:11']} with initial datum $g(x) = \min(|x|,|x-\frac{1}{2}|-\frac{1}{4})$ for $x\in \mathbb{R}$. Here, $\Omega = [-\frac{5}{2},\frac{5}{2}]$, $T = 1$ for (A)--(D), and $\Omega = [-\frac{11}{2},\frac{11}{2}]$, $T = \frac{1}{2}$ for (E)--(F).
• Figure 6.1: Approximation of $\overline H(p)$ at $p = (3,1)$ for Example \ref{['ex:Hbar1']}.
• Figure 6.2: Approximation of $\overline H$ for Example \ref{['ex:Hbar2']}.

Theorems & Definitions (33)

• Definition 1: Effective Hamiltonian
• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• proof : Proof of \ref{['upperbd']}
• proof : Proof of \ref{['lowerbd']}
• Proposition 4.1
• proof
• proof : Proof of Theorem \ref{['thm:new']}