Homogenization of first order equations with $u/\epsilon$-periodic Hamiltonian: Rate of convergence as $\epsilon\to 0$ and numerical approximation of the effective Hamiltonian

TL;DR

This paper investigates the convergence rate of solutions to homogenized first order Hamilton-Jacobi equations with a unique $u^rac{ ext{epsilon}}{ ext{epsilon}}$ periodic dependence and develops numerical schemes for approximating the effective Hamiltonian.

Contribution

It establishes the convergence rate for these nonstandard homogenization problems and proposes Eulerian schemes for numerical approximation of the effective Hamiltonian.

Findings

Convergence rates match those of standard problems without $u^rac{ ext{epsilon}}{ ext{epsilon}}$ dependence.

Eulerian schemes converge to the effective Hamiltonian as grid steps tend to zero.

Numerical methods effectively approximate the nonstandard cell problems.

Abstract

We consider homogenization problems for first order Hamilton-Jacobi equations with $u^\epsilon/\epsilon$ periodic dependence, recently introduced by C. Imbert and R. Monneau, and also studied by G. Barles: this unusual dependence leads to a nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter $\epsilon$ tends to 0. We obtain the same rates as those obtained by I. Capuzzo Dolcetta and H. Ishii for the more usual homogenization problems without the dependence in $u^\epsilon/\epsilon$. In a second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.