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Published December 2004 | Published
Journal Article Open

Multiscale Finite Element Methods for Nonlinear Problems and their Applications

Abstract

In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.

Additional Information

© 2004 International Press. Received: June 22, 2004; accepted (in revised version): September 28, 2004. Communicated by Shi Jin. The research of Y. E. is partially supported by NSF grants DMS-0327713 and EIA-0218229. The research of T.Y.H. is partially supported by the NSF ITR grant ACI-0204932. We would like to acknowledge anonymous reviewers for their helpful comments which helped to improve the quality of the paper.

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