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arXiv · 2009.11259

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

Timo Sprekeler·Hung V. Tran
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Abstract

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 optimal gradient and Hessian bounds with correction terms taken into account in the $L^p$-norm. 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.

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Timo Sprekeler, Hung V. Tran. 2020-09-23. Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations. https://doi.org/10.1137/20m137121x

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