NDLI: Convergence of a Nonconforming Multiscale Finite Element Method

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Hou, Thomas Y. Efendiev, Yalchin R. Wu, Xiao-Hui
Copyright Year	2000
Abstract	The multiscale finite element method (MsFEM) [T. Y. Hou, X. H. Wu, and Z. Cai, Math. Comp., 1998, to appear; T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169--189] has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients. This is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator. Our previous study reveals that the leading order error in this approach is caused by the ``resonant sampling,' which leads to large error when the mesh size is close to the small scale of the continuous problem. Similar difficulty also arises in numerical upscaling methods. An oversampling technique has been introduced to alleviate this difficulty [T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169--189]. A consequence of the oversampling method is that the resulting finite element method is no longer conforming. Here we give a detailed analysis of the nonconforming error. Our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale. We show that the cell resonance error is of lower order. Our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice.
Starting Page	888
Ending Page	910
Page Count	23
File Format	PDF
ISSN	00361429
DOI	10.1137/S0036142997330329
e-ISSN	10957170
Issue Number	3
Volume Number	37
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-25
Access Restriction	Subscribed
Subject Keyword	homogenization Other matrix algorithms finite element Iterative methods for linear systems multiscale resonant sampling
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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