Geodesic
On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1345-1380

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We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the equation can be incomplete. A theoretical foundation of the approach in the limit of infinitely small oscillations of the coefficients is provided, using the classical theory of homogenization. We present a comprehensive study of the implementation aspects of our method, and a set of numerical tests and comparisons that show the potential practical interest of the approach. The approach detailed in this article improves on an earlier version briefly presented in [C. Le Bris, F. Legoll and K. Li, C.R. Acad. Sci. Paris, Série I  351 (2013) 265–270].

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017061
Classification : 35J, 35B27, 74Q15
Keywords: Elliptic PDEs, Oscillatory coefficients, Homogenization, Coarse-graining

Le Bris, Claude 1 ; Legoll, Frédéric 1 ; Lemaire, Simon 1

1 
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     title = {On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1345--1380},
     publisher = {EDP-Sciences},
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Le Bris, Claude; Legoll, Frédéric; Lemaire, Simon. On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1345-1380. doi: 10.1051/cocv/2017061

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