Convergence of a nonconforming multiscale finite element method

Yalchin R. Efendiev*, Thomas Y. Hou, Xiao Hui Wu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

197 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)888-910
Number of pages23
JournalSIAM Journal on Numerical Analysis
Volume37
Issue number3
DOIs
StatePublished - Jan 1 2000

Keywords

  • Finite element
  • Homogenization
  • Multiscale
  • Resonant sampling

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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