Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form

Abstract

We characterize diffusion matrices that yield a $L^{\infty }$ convergence rate of $\mathcal{O}({\epsilon }^2)$ in the theory of periodic homogenization of linear elliptic equations in nondivergence-form. Such type-${\epsilon }^2$ diffusion matrices are of particular interest as the optimal rate of convergence in the generic case is only $\mathcal{O}(\epsilon )$. First, we provide a new class of type-${\epsilon }^2$ diffusion matrices, confirming a conjecture posed in [15]. Then, we give a complete characterization of diagonal diffusion matrices in two dimensions and a systematic study in higher dimensions.

arXiv comment31 pages; added Section 3.3

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Cited by in corpus (2)

• Finite element approximation of stationary Fokker--Planck--Kolmogorov equations with application to periodic numerical homogenization
• Optimal convergence rates in stochastic homogenization in a balanced random environment