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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

A convergence analysis of Generalized Multiscale Finite Element Methods

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Author(s):
Abreu, Eduardo [1] ; Diaz, Ciro [1] ; Galvis, Juan [2]
Total Authors: 3
Affiliation:
[1] Univ Estadual Campinas, UNICAMP, Dept Appl Math Univ, BR-13083970 Campinas, SP - Brazil
[2] Univ Nacl Colombia, Dept Matemat, Carrera 45 26-85, Edificio Uriel Gutierrez, Bogota - Colombia
Total Affiliations: 2
Document type: Journal article
Source: Journal of Computational Physics; v. 396, p. 303-324, NOV 1 2019.
Web of Science Citations: 0
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. (C) 2019 Elsevier Inc. All rights reserved. (AU)

FAPESP's process: 16/23374-1 - Conservation laws, balance laws and related PDEs with discontinuous and nonlocal fluxes in applied sciences: numerical analysis, theory and applications
Grantee:Eduardo Cardoso de Abreu
Support type: Regular Research Grants