A convergence analysis of Generalized Multiscale Finite Element Methods

Autor(es)

Abreu, Eduardo ^[1] ; Diaz, Ciro ^[1] ; Galvis, Juan ^[2]

Afiliação do(s) autor(es)

^[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

• Tipo de documento: Artigo Científico
• Fonte: Journal of Computational Physics; v. 396, p. 303-324, NOV 1 2019.
• Citações Web of Science: 0

Resumo

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)

• Processo FAPESP: 16/23374-1 - Leis de conservação, leis de equilíbrio e EDPs relacionadas com fluxos descontínuos e não-locais em ciências aplicadas: análise numérica, teoria e aplicações
• Beneficiário: Eduardo Cardoso de Abreu
• Modalidade de apoio: Auxílio à Pesquisa - Regular