dc.contributor.author Efendiev, Yalchin R. dc.contributor.author Galvis, Juan dc.contributor.author Wu, Xiao-Hui dc.date.accessioned 2016-02-25T13:43:40Z dc.date.available 2016-02-25T13:43:40Z dc.date.issued 2011-02 dc.identifier.citation Efendiev Y, Galvis J, Wu X-H (2011) Multiscale finite element methods for high-contrast problems using local spectral basis functions. Journal of Computational Physics 230: 937–955. Available: http://dx.doi.org/10.1016/j.jcp.2010.09.026. dc.identifier.issn 0021-9991 dc.identifier.doi 10.1016/j.jcp.2010.09.026 dc.identifier.uri http://hdl.handle.net/10754/598916 dc.description.abstract In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/Λ*)1/2, where Λ* is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings. © 2010. dc.description.sponsorship The work of Y.E. and J.G. is partially supported by Award Number KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). Y.E.'s research is partially supported by NSF (0724704, 0811180, 0934837) and DOE. We would like to thank the anonymous reviewers for their suggestions that helped to improve the paper. dc.publisher Elsevier BV dc.subject High contrast dc.subject Multiscale finite element dc.subject Porous media dc.subject Spectral dc.title Multiscale finite element methods for high-contrast problems using local spectral basis functions dc.type Article dc.identifier.journal Journal of Computational Physics dc.contributor.institution Texas A and M University, College Station, United States dc.contributor.institution ExxonMobil, Irving, United States kaust.grant.number KUS-C1-016-04
﻿