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

TL;DR

This paper develops a method to approximate elliptic equations with oscillatory coefficients by constant coefficient problems, providing theoretical justification and numerical validation, with practical engineering applications in incomplete data scenarios.

Contribution

It introduces a new approach to approximate oscillatory coefficients with a constant matrix, supported by homogenization theory and practical implementation insights.

Findings

The method effectively approximates oscillatory coefficients in elliptic problems.

Numerical tests demonstrate the approach's accuracy and potential for engineering applications.

The approach improves upon previous methods with better practical and theoretical foundations.

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].