Optimal Convergence Rates for Elliptic Homogenization Problems in Nondivergence-Form: Analysis and Numerical Illustrations
Timo Sprekeler, Hung V. Tran · Multiscale Modeling and Simulation · 2021
We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for the convergence of $u^{\varepsilon}$ to the solution of the corresponding homogenized problem in the $W^{1,p}$-norm is $\mathcal{O}(\varepsilon)$. We further obtain gradient and Hessian bounds with correction terms taken into account in the $L^p$-norm and recover the known optimal convergence rate in the $L^{\infty}$-norm under weak assumptions on the data. We then provide an explicit $c$-bad diffusion matrix and use it to perform various numerical experiments, which demonstrate the optimality of the obtained rates. Finally, we discuss extensions of the results to the case of nonsmooth domains and their utility in regard to numerical homogenization.