Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915031847796736 |
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| author | Qian, Jianliang Sprekeler, Timo Tran, Hung V. Yu, Yifeng |
| author_facet | Qian, Jianliang Sprekeler, Timo Tran, Hung V. Yu, Yifeng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03091 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations Qian, Jianliang Sprekeler, Timo Tran, Hung V. Yu, Yifeng Analysis of PDEs Numerical Analysis 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. |
| title | Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2402.03091 |