A Multiscale Model Reduction Method for Partial Differential Equations

Abstract

We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is in H^2, and can be approximated by a regular coarse mesh. When the multiscale problem has scale separation and a periodic structure, our method recovers the traditional homogenized equation. Furthermore, we provide error analysis for our method and show that the solution to the effective equation is close to the original multiscale solution in the H^1 norm. Numerical results are presented to demonstrate the accuracy and robustness of the proposed method for several multiscale problems without scale separation, including a problem with a high contrast coefficient.

Additional Information

© 2014 EDP Sciences, SMAI. Received August 3, 2013; Published online February 20, 2014. This research was supported in part by a DOE Grant DE-FG02-06ER25727, a AFOSR MURI Grant FA9550-09-1-0613 and an NSF FRG Grant DMS-1159138. The research of Dr. Z. Shi was in part supported by a NSFC Grant 11201257.

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