Geodesic
Operator Error Estimates in $L_2$ for Homogenization of an Elliptic Dirichlet Problem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 3, pp. 91-96

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In a bounded domain ${\mathcal O} \subset {\mathbb R}^d$ with $C^{1,1}$ boundary a matrix elliptic second-order operator ${A}_{D,\varepsilon}$ with Dirichlet boundary condition is studied. The coefficients of this operator are periodic and depend on $\mathbf{x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. The sharp-order error estimate $\|{A}_{D,\varepsilon}^{-1} - ({A}_D^0)^{-1} \|_{L_2 \to L_2} \le C \varepsilon$ is obtained. Here ${A}^0_D$ is an effective operator with constant coefficients and Dirichlet boundary condition.
Keywords: periodic differential operators, homogenization, effective operator, operator error estimates. 
@article{FAA_2012_46_3_a8,
     author = {T. A. Suslina},
     title = {Operator {Error} {Estimates} in $L_2$ for {Homogenization} of an {Elliptic} {Dirichlet} {Problem}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {91--96},
     publisher = {mathdoc},
     volume = {46},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2012_46_3_a8/}
}
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T. A. Suslina. Operator Error Estimates in $L_2$ for Homogenization of an Elliptic Dirichlet Problem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 3, pp. 91-96. http://geodesic.mathdoc.fr/item/FAA_2012_46_3_a8/