A Multiscale Enrichment Procedure for Nonlinear Monotone Operators

¹ 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
² 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
³ Departmento de matematics, Universidad Nacional de Colombia, Bogota D.C., Colombia
jcgalvisa@unal.edu.co

Received: 29 July 2013

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.