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.
Mots-clés : generalized multiscale finite element method, nonlinear equations, high-contrast
@article{M2AN_2014__48_2_475_0,
author = {Efendiev, Y. and Galvis, J. and Presho, M. and Zhou, J.},
title = {A {Multiscale} {Enrichment} {Procedure} for {Nonlinear} {Monotone} {Operators}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {475--491},
publisher = {EDP-Sciences},
volume = {48},
number = {2},
year = {2014},
doi = {10.1051/m2an/2013116},
mrnumber = {3177854},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013116/}
}
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 PY - 2014 SP - 475 EP - 491 VL - 48 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013116/ DO - 10.1051/m2an/2013116 LA - en ID - M2AN_2014__48_2_475_0 ER -
%0 Journal Article %A Efendiev, Y. %A Galvis, J. %A Presho, M. %A Zhou, J. %T A Multiscale Enrichment Procedure for Nonlinear Monotone Operators %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 475-491 %V 48 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013116/ %R 10.1051/m2an/2013116 %G en %F M2AN_2014__48_2_475_0
Efendiev, Y.; Galvis, J.; Presho, M.; Zhou, J. A Multiscale Enrichment Procedure for Nonlinear Monotone Operators. ESAIM: Mathematical Modelling and Numerical Analysis , Multiscale problems and techniques. Special Issue, Tome 48 (2014) no. 2, pp. 475-491. doi : 10.1051/m2an/2013116. http://www.numdam.org/articles/10.1051/m2an/2013116/
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