dc.contributor.author Calo, Victor dc.contributor.author Efendiev, Y. dc.contributor.author Galvis, J. dc.contributor.author Ghommem, M. dc.date.accessioned 2017-03-24T11:54:12Z dc.date.available 2017-03-24T11:54:12Z dc.date.created 2017-03-23T06:59:54Z dc.date.issued 2014 dc.identifier.citation Calo, V. and Efendiev, Y. and Galvis, J. and Ghommem, M. 2014. Multiscale empirical interpolation for solving nonlinear PDEs. Journal of Computational Physics. 278 (1): pp. 204-220. dc.identifier.uri http://hdl.handle.net/20.500.11937/51608 dc.identifier.doi 10.1016/j.jcp.2014.07.052 dc.description.abstract © 2014 Elsevier Inc.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 system residuals and Jacobians on the fine grid. We use empirical interpolation concepts to evaluate these residuals and Jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse-scale problem rather than the fully-resolved fine scale one. The empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times. The coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system. 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 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 and significant computational cost reduction. dc.publisher Academic Press dc.title Multiscale empirical interpolation for solving nonlinear PDEs dc.type Journal Article dcterms.source.volume 278 dcterms.source.number 1 dcterms.source.startPage 204 dcterms.source.endPage 220 dcterms.source.issn 0021-9991 dcterms.source.title Journal of Computational Physics curtin.department Department of Applied Geology curtin.accessStatus Fulltext not available
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