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.

31 pages; added Section 3.3

References in corpus (3)

• Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization
• Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations
• Discontinuous Galerkin and $C^0$-IP finite element approximation of periodic Hamilton--Jacobi--Bellman--Isaacs problems with application to numerical homogenization

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