Homogenization of the elliptic Dirichlet problem: operator error estimates in $L_2$

TL;DR

This paper establishes a sharp order error estimate in the L2 norm for the homogenization of elliptic operators with Dirichlet boundary conditions on a smooth domain, quantifying the approximation accuracy of the effective operator.

Contribution

The paper provides a precise operator error estimate of order epsilon for the homogenization of matrix elliptic operators with Dirichlet boundary conditions.

Findings

Operator error estimate of order epsilon in L2 norm

Effective operator with constant coefficients approximates the original operator

Results applicable to bounded domains with C^2 boundary

Abstract

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the Dirichlet boundary condition. Here $\varepsilon>0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf{x}/\varepsilon$. A sharp order operator error estimate $\|\mathcal{A}_{D,\varepsilon}^{-1} - (\mathcal{A}_D^0)^{-1} \|_{L_2 \to L_2} \leq C \varepsilon$ is obtained. Here $\mathcal{A}^0_D$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.