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

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