A Multiscale Enrichment Procedure for Nonlinear Monotone Operators

Y. Efendiev; J. Galvis; M. Presho; J. Zhou

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 2, page 475-491
  • ISSN: 0764-583X

Abstract

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In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461–1483.] is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.

How to cite

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Efendiev, Y., et al. "A Multiscale Enrichment Procedure for Nonlinear Monotone Operators." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.2 (2014): 475-491. <http://eudml.org/doc/273142>.

@article{Efendiev2014,
abstract = {In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461–1483.] is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.},
author = {Efendiev, Y., Galvis, J., Presho, M., Zhou, J.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {generalized multiscale finite element method; nonlinear equations; high-contrast; domain decomposition; nonlinear elliptic problems; nonlinear eigenvalue problem; numerical example},
language = {eng},
number = {2},
pages = {475-491},
publisher = {EDP-Sciences},
title = {A Multiscale Enrichment Procedure for Nonlinear Monotone Operators},
url = {http://eudml.org/doc/273142},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Efendiev, Y.
AU - Galvis, J.
AU - Presho, M.
AU - Zhou, J.
TI - A Multiscale Enrichment Procedure for Nonlinear Monotone Operators
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 2
SP - 475
EP - 491
AB - In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461–1483.] is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.
LA - eng
KW - generalized multiscale finite element method; nonlinear equations; high-contrast; domain decomposition; nonlinear elliptic problems; nonlinear eigenvalue problem; numerical example
UR - http://eudml.org/doc/273142
ER -

References

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