A Multiscale Enrichment Procedure for Nonlinear Monotone Operators

Yalchin R. Efendiev, J. Galvis, M. Presho, J. Zhou

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)475-491
Number of pages17
JournalESAIM: Mathematical Modelling and Numerical Analysis
Volume48
Issue number2
DOIs
StatePublished - Mar 11 2014

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