Metric based up-scaling

Creators: Owhadi, Houman; Zhang, Lei

Abstract

We consider divergence form elliptic operators in dimension n ≥ 2 with L∞ coefficients. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators.

Additional Information

(Submitted on 11 May 2005 (v1), last revised 16 Nov 2005 (this version, v5)). November 15, 2005. Part of the work of the first author has been supported by CNRS. The authors would like to thank Jean-Michel Roquejoffre for indicating us the correct references on nonlinear PDEs, Mathieu Desbrun for enlightening discussions on discrete exterior calculus [58] (a powerful tool that has put into evidence the intrinsic way to define discrete differential operators on irregular triangulations), Tom Hou and Jerry Marsden for stimulating discussions on multi-scale computation, Clothilde Melot and Stéphane Jaffard for stimulating discussions on multi-fractal analysis. Thanks are also due to Lexing Ying and Laurent Demanet for useful comments on the manuscript and G. Ben Arous for indicating us reference [72]. Many thanks are also due to Stefan Müller (MPI, Leipzig) for valuable suggestions and for indicating us the Hierarchical Matrices methods. We would like also to thank G. Allaire, F. Murat and S.R.S. Varadhan for stimulating discussions at the CIRM workshop on random homogenization and P. Schröder for stimulating discussions on splines based methods. We also thank an anonymous referee for detailed comments and suggestions.

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