On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators

2016/12/17 by Claude Le Bris, Bris, Claude Le, Frédéric Legoll +3 · 1 citation
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Composite Material Mechanics #FOS: Mathematics #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.1612.05807

openalex publication_date 2016/12/17 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28

Abstract

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 2013].

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