A convergence analysis of Generalized Multiscale Finite Element Methods

Eduardo Abreu, Ciro Díaz, Juan Galvis

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of Y. Efendiev et al. (2011) [22]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis.
Original languageEnglish (US)
Pages (from-to)303-324
Number of pages22
JournalJournal of Computational Physics
Volume396
DOIs
StatePublished - Jul 24 2019
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-10
Acknowledgements: Eduardo Abreu thanks the FAPESP for support under grant 2016/23374-1. Juan Galvis wants to thank KAUST hospitality where part of this work was developed and also the discussion on coarse space approximations properties and related topics with several colleagues, among them, Joerg Willems, Marcus Sarkis, Raytcho Lazarov, Jhonny Guzmán, Chia-Chieh Chu, Florian Maris, Yalchin Efendiev and Guanglian Li.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)
  • Computer Science Applications

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