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

Abstract

We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\epsilon ):D^2{u}^{\epsilon }=f$ subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for the convergence of $u^{\epsilon }$ to the solution of the corresponding homogenized problem in the $W^{1,p}$-norm is $\mathcal{O}(\epsilon )$. 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.

arXiv comment20 pages; added Section 4.2

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Cited by in corpus (5)

• Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization
• Discontinuous Galerkin and $C^0$-IP finite element approximation of periodic Hamilton--Jacobi--Bellman--Isaacs problems with application to numerical homogenization
• Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form
• Computational multiscale methods for nondivergence-form elliptic partial differential equations
• Optimal convergence rates in stochastic homogenization in a balanced random environment