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

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 ${\mathcal{A}}_{D,\epsilon }$ with the Dirichlet boundary condition. Here $\epsilon >0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf{x}/\epsilon$. A sharp order operator error estimate $\Vert {\mathcal{A}}_{D,\epsilon }^{-1}-({\mathcal{A}}_D^0{)}^{-1}{\Vert }_{L_2\to L_2}\le C\epsilon$ is obtained. Here ${\mathcal{A}}_D^0$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.

13 pages

References in corpus (1)

• Interior error estimate for periodic homogenization

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