Multiscale Empirical Interpolation for Solving Nonlinear PDEs using   Generalized Multiscale Finite Element Methods
  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. 
Victor Calo, Yalchin Efendiev, Juan Galvis, Mehdi Ghommem
math.NA