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In this paper, we propose a multiscale empirical interpolation  method for solving nonlinear multiscale partial differential  equations. The proposed method combines empirical interpolation  techniques and local multiscale methods, such as the Generalized  Multiscale Finite Element Method (GMsFEM). To solve nonlinear  equations, the GMsFEM is used to represent the solution on a coarse  grid with multiscale basis functions computed offline. Computing  the GMsFEM solution involves calculating the residuals on the fine  grid. We use empirical interpolation concepts to evaluate the  residuals and the Jacobians of the multiscale system with a  computational cost which is proportional to the coarse scale problem  rather than the fully-resolved fine scale one. Empirical  interpolation methods use basis functions and an inexpensive  inversion which are computed in the offline stage for finding the  coefficients in the expansion based on a limited number of nonlinear  function evaluations. The proposed multiscale empirical  interpolation techniques: (1) divide computing the nonlinear  function into coarse regions; (2) evaluate contributions of  nonlinear functions in each coarse region taking advantage of a  reduced-order representation of the solution; and (3) introduce  multiscale proper-orthogonal-decomposition techniques to find  appropriate interpolation vectors. We demonstrate the effectiveness  of the proposed methods on several examples of nonlinear multiscale  PDEs that are solved with Newton's methods and fully-implicit time  marching schemes. Our numerical results show that the proposed  methods provide a robust framework for solving nonlinear multiscale  PDEs on a coarse grid with bounded error.

Multiscale Empirical Interpolation for Solving Nonlinear PDEs using Generalized Multiscale Finite Element Methods