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

• Published in: arXiv.org
• Main Authors: Sprekeler, Timo, Tran, Hung V
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 06.10.2020
• Subjects: Boundary conditions; Convergence; Dirichlet problem; Elliptic functions; Homogenization; Mathematics - Analysis of PDEs; Optimization
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2009.11259

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.