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.

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.

DOI : 10.1051/m2an/2013116
Classification : 35J60, 65N30
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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/m2an/2013116/

[1] J. Aarnes, S. Krogstad and K. Lie, 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. | MR | Zbl

[2] T. Arbogast, G. Pencheva, M. Wheeler and I. Yotov, A multiscale mortar mixed finite element method. SIAM Multiscale Model. Simul. 6 (2007) 319-346. | MR

[3] X. Cai and D. Keyes, Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Sci. Comput. 24 (2002) 183-200. | MR | Zbl

[4] X. Chen and Y. Lou, 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. | MR | Zbl

[5] M. Dryja and W. Hackbusch, On the nonlinear domain decomposition method. BIT 37 (1997) 296-311. | MR | Zbl

[6] 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. Comput. Method Appl. Math. 12 (2012) 1-22. | MR | Zbl

[7] Y. Efendiev, J. Galvis and T. Hou, Generalized Multiscale Finite Element Method. J. Comput. Phys. (2013) 116-135. | MR

[8] Y. Efendiev, J. Galvis, G. Li and M. Presho, Generalized Multiscale Finite Element Methods. Oversampling strategies. To appear in Int. J. Multiscale Comput. Engrg.

[9] Y. Efendiev, J. Galvis and X. Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230 (2011) 937-955. | MR

[10] J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media. SIAM Multiscale Model. Simul. 8 (2010) 1461-1483. | MR | Zbl

[11] Y. Efendiev and T. Hou, Multiscale Finite Element Methods: Theory and Applications. Springer, New York (2009). | MR | Zbl

[12] T. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR | Zbl

[13] T. Hughes, G. Feijóo, L. Mazzei and J. Quincy, The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24. | MR | Zbl

[14] P. Jenny, S. Lee and H. Tchelepi, Multi-scale finite volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47-67. | Zbl

[15] T. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. BIT 37 (1997) 296-311. | Zbl

[16] P. Solin and S. Giani, An iterative adaptive finite element method for elliptic eigenvalue problems. J. Comput. Appl. Math. 236 (2012) 4582-4599 | MR | Zbl

[17] X. Tai and M. Espedal, Rate of convergence of some space decomposition methods for linear and nonlinear problems. Springer-Verlag, Berlin-Heidelburg (2008). | Zbl

[18] J. Xu and L. Zikatanov, On an energy minimizing basis for algebraic multigrid methods. Comput. Visual Sci. 7 (2004) 121-127. | MR | Zbl

[19] X. Yao and J. Zhou, Numerical methods for computing nonlinear eigenpairs. Part I. Iso-homogeneous cases. SIAM J. Sci. Comput. 29 (2007) 1355-1374. | MR | Zbl

[20] X. Yao and J. Zhou, Numerical methods for computing nonlinear eigenpairs. Part II. Non iso-homogenous cases. SIAM J. Sci. Comp. 30 (2008) 937-956. | MR | Zbl

[21] E. Zeidler, Nonlinear Functional Analysis and Its Applications III. Springer-Verlag, New York (1985). | MR | Zbl

Cité par Sources :