Geodesic
Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 2, pp. 92-96 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain with boundary of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, consider a matrix elliptic second-order differential operator $A_{D,\varepsilon}$ with Dirichlet boundary condition. Here $\varepsilon >\nobreak0$ is a small parameter; the coefficients of $A_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. The operator $A_{D,\varepsilon}^{-1}$ in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ is approximated with an error of order $\varepsilon^{1/2}$. The approximation is given by the sum of the operator $(A^0_D)^{-1}$ and a first-order corrector. Here $A^0_D$ is an effective operator with constant coefficients and Dirichlet boundary condition.
Keywords: homogenization of periodic differential operators, effective operator, corrector, operator error estimates. 
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     author = {M. A. Pakhnin and T. A. Suslina},
     title = {Homogenization of the {Elliptic} {Dirichlet} {Problem:} {Error} {Estimates} in the $(L_2\to H^1)${-Norm}},
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M. A. Pakhnin; T. A. Suslina. Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 2, pp. 92-96. http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a9/

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