Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T04:45:26.225Z Has data issue: false hasContentIssue false
  • English
  • Français

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. [email protected]; [email protected]; [email protected] SRI-Center for Numerical Porous Media, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia; [email protected]
J. Galvis
Affiliation:
Departmento de matematics, Universidad Nacional de Colombia, Bogota D.C., Colombia; [email protected]
M. Presho
Affiliation:
Department of Mathematics and Institute for Scientific Computation, Texas A & M University, College Station, TX 77843, USA. [email protected]; [email protected]; [email protected]
J. Zhou
Affiliation:
Department of Mathematics and Institute for Scientific Computation, Texas A & M University, College Station, TX 77843, USA. [email protected]; [email protected]; [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aarnes, J., Krogstad, S. and Lie, K., A hierarchical multiscale method for two-phase flow based on upon mixed finite elements and nonuniform coarse grids. SIAM Multiscale Model. Simul. 5 (2006) 337–363. Google Scholar
Arbogast, T., Pencheva, G., Wheeler, M. and Yotov, I., A multiscale mortar mixed finite element method. SIAM Multiscale Model. Simul. 6 (2007) 319–346. Google Scholar
Cai, X. and Keyes, D., Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Sci. Comput. 24 (2002) 183–200. Google Scholar
Chen, X. and Lou, Y., Principal eigenvalue and eigenfunction of elliptic operator with large convection and its application to a competition model. Indiana Univ. Math. J. 57 (2008) 627–658. Google Scholar
Dryja, M. and Hackbusch, W., On the nonlinear domain decomposition method. BIT 37 (1997) 296–311. Google Scholar
Efendiev, Y., Galvis, J., Lazarov, R., Margenov, S. and Ren, J., Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Comput. Method Appl. Math. 12 (2012) 1–22. Google Scholar
Efendiev, Y., Galvis, J. and Hou, T., Generalized Multiscale Finite Element Method. J. Comput. Phys. (2013) 116–135. Google Scholar
Y. Efendiev, J. Galvis, G. Li and M. Presho, Generalized Multiscale Finite Element Methods. Oversampling strategies. To appear in Int. J. Multiscale Comput. Engrg.
Efendiev, Y., Galvis, J. and Wu, X., Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230 (2011) 937–955. Google Scholar
Galvis, J. and Efendiev, Y., Domain decomposition preconditioners for multiscale flows in high contrast media. SIAM Multiscale Model. Simul. 8 (2010) 1461–1483. Google Scholar
Y. Efendiev and T. Hou, Multiscale Finite Element Methods: Theory and Applications. Springer, New York (2009).
Hou, T. and Wu, X., A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. Google Scholar
Hughes, T., Feijóo, G., Mazzei, L. and Quincy, J., The variational multiscale method – a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3–24. Google Scholar
Jenny, P., Lee, S. and Tchelepi, H., Multi-scale finite volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47–67. Google Scholar
Mathew, T., Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. BIT 37 (1997) 296–311. Google Scholar
Solin, P. and Giani, S., An iterative adaptive finite element method for elliptic eigenvalue problems. J. Comput. Appl. Math. 236 (2012) 4582–4599 Google Scholar
X. Tai and M. Espedal, Rate of convergence of some space decomposition methods for linear and nonlinear problems. Springer-Verlag, Berlin-Heidelburg (2008).
Xu, J. and Zikatanov, L., On an energy minimizing basis for algebraic multigrid methods. Comput. Visual Sci. 7 (2004) 121–127. Google Scholar
Yao, X. and Zhou, J., Numerical methods for computing nonlinear eigenpairs. Part I. Iso-homogeneous cases. SIAM J. Sci. Comput. 29 (2007) 1355–1374. Google Scholar
Yao, X. and Zhou, J., Numerical methods for computing nonlinear eigenpairs. Part II. Non iso-homogenous cases. SIAM J. Sci. Comp. 30 (2008) 937–956. Google Scholar
E. Zeidler, Nonlinear Functional Analysis and Its Applications III. Springer-Verlag, New York (1985).