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Multiscale Finite Element Methods for Nonlinear Problems and Their Applications

Abstract. In this paper we propose a generalization of multiscale finite element methods (MsFEM) 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.

Key words. multiscale, finite element, upscaling, nonlinear, elliptic, oversampling

AMS Classification number. 65N99