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We study optimal convergence rates in the periodic homogenization of linearelliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$subject to a homogeneous Dirichlet boundary condition. We show that the optimalrate for the convergence of $u^{\varepsilon}$ to the solution of thecorresponding homogenized problem in the $W^{1,p}$-norm is$\mathcal{O}(\varepsilon)$. We further obtain optimal gradient and Hessianbounds with correction terms taken into account in the $L^p$-norm. We thenprovide an explicit $c$-bad diffusion matrix and use it to perform variousnumerical experiments, which demonstrate the optimality of the obtained rates.

Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations