Abstract
We use uniform W2,p estimates to obtain corrector results for periodic homogenization problems of the form A(x/ε):D2uε = f subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.
Highlights
In this work we consider second-order elliptic equations of nondivergence structure, involving rapidly oscillating coefficients, of the form A (︁ · ε )︁ : D2uε := n ∑︁ aij
The main goal of this paper is to propose and analyze a numerical homogenization scheme for (1.1), (1.2) that is based on finite element approximations
An a priori error analysis for the fully discrete finite element heterogeneous multiscale method (HMM) for elliptic homogenization problems in divergence-form can be found in the work [1] by Abdulle
Summary
In this work we consider second-order elliptic equations of nondivergence structure, involving rapidly oscillating coefficients, of the form. Homogenization, nondivergence-form elliptic PDE, finite element methods. It is important to note that if A is sufficiently smooth, equation (1.1) can be rewritten in divergenceform, This equation does not fit into the framework of divergence-form homogenization problems such as (1.5), because of the ε−1 term in front of the first-order term in (1.6). An a priori error analysis for the fully discrete finite element HMM for elliptic homogenization problems in divergence-form can be found in the work [1] by Abdulle. The first step in the development of the proposed numerical homogenization scheme is the construction of a finite element method to obtain approximations (mh)h>0 ⊂ Hp1er(Y ) to the invariant measure with optimal order convergence rate.
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