dc.contributor.author Calo, Victor M. dc.contributor.author Efendiev, Yalchin R. dc.contributor.author Galvis, Juan dc.contributor.author Ghommem, Mehdi dc.date.accessioned 2015-08-03T12:18:24Z dc.date.available 2015-08-03T12:18:24Z dc.date.issued 2014-12 dc.identifier.issn 00219991 dc.identifier.doi 10.1016/j.jcp.2014.07.052 dc.identifier.uri http://hdl.handle.net/10754/563887 dc.description.abstract 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.description.sponsorship YE's work is supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and by FA9550-11-1-0341 from the Air Force Office of Scientific Research. dc.publisher Elsevier BV dc.relation.url http://arxiv.org/abs/arXiv:1407.0103v1 dc.subject Discrete empirical interpolation method dc.subject Generalized multiscale finite element methods dc.subject Model reduction dc.subject Nonlinear PDEs dc.title Multiscale empirical interpolation for solving nonlinear PDEs dc.type Article dc.contributor.department Numerical Porous Media SRI Center (NumPor) dc.contributor.department Applied Mathematics and Computational Science Program dc.contributor.department Earth Science and Engineering Program dc.contributor.department Physical Sciences and Engineering (PSE) Division dc.contributor.department Environmental Science and Engineering Program dc.contributor.department Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division dc.identifier.journal Journal of Computational Physics dc.contributor.institution Department of Mathematics and Institute for Scientific Computation (ISC), Texas AandM UniversityCollege Station, TX, United States dc.contributor.institution Departamento de Matemáticas, Universidad Nacional de Colombia, Carrera 45 No 26-85 Edificio Uriel GutierrézBogotá D.C., Colombia dc.identifier.arxivid arXiv:1407.0103 kaust.person Calo, Victor M. kaust.person Efendiev, Yalchin R. kaust.person Ghommem, Mehdi
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