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A Multiscale Enrichment Procedure for Nonlinear Monotone Operators

Published online by Cambridge University Press:  11 March 2014

Y. Efendiev ,
J. Galvis ,
M. Presho  and
J. Zhou
Y. Efendiev
Affiliation:
Department of Mathematics and Institute for Scientific Computation, Texas A & M University, College Station, TX 77843, USA. efendiev@math.tamu.edu; mpresho@math.tamu.edu; jzhou@math.tamu.edu SRI-Center for Numerical Porous Media, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia; yalchin.efendiev@kaust.edu.sa
J. Galvis
Affiliation:
Departmento de matematics, Universidad Nacional de Colombia, Bogota D.C., Colombia; jcgalvisa@unal.edu.co
M. Presho
Affiliation:
Department of Mathematics and Institute for Scientific Computation, Texas A & M University, College Station, TX 77843, USA. efendiev@math.tamu.edu; mpresho@math.tamu.edu; jzhou@math.tamu.edu
J. Zhou
Affiliation:
Department of Mathematics and Institute for Scientific Computation, Texas A & M University, College Station, TX 77843, USA. efendiev@math.tamu.edu; mpresho@math.tamu.edu; jzhou@math.tamu.edu
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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.

Keywords

Generalized multiscale finite element methodnonlinear equationshigh-contrast
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 475 - 491
Copyright
© EDP Sciences, SMAI, 2014

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