2017, 12(4): 619-642. doi: 10.3934/nhm.2017025

A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media

1. 

Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, SAR, China

2. 

Department of Mathematics, Texas A & M University, College Station, TX 77843, USA

3. 

Department of Mathematics & Statistics, Old Dominion University, Norfolk, VA 23529, USA

Received  January 2016 Revised  October 2016 Published  October 2017

Fund Project: Eric Chung would like to thank the partial support of the CUHK Direct Grant for Research 2015/16 and the Hong Kong RGC General Research Fund (Project: 14317516). YE would like to thank the partial support from NSF 1620318, the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165, the mega-grant of the Russian Federation Government (N 14.Y26.31.0013), and National Priorities Research Program grant NPRP grant 7-1482-1278 from the Qatar National Research Fund

In this paper, we present a multiscale model reduction framework within Generalized Multiscale Finite Element Method (GMsFEM) for nonlinear elliptic problems. We consider an exemplary problem, which consists of nonlinear p-Laplacian with heterogeneous coefficients. The main challenging feature of this problem is that local subgrid models are nonlinear involving the gradient of the solution (e.g., in the case of scale separation, when using homogenization). Our main objective is to develop snapshots and local spectral problems, which are the main ingredients of GMsFEM, for these problems. Our contributions can be summarized as follows. (1) We re-cast the multiscale model reduction problem onto the boundaries of coarse cells. This is important and allows capturing separable scales as discussed. (2) We introduce nonlinear eigenvalue problems in the snapshot space for these nonlinear "harmonic" functions. (3) We present convergence analysis and numerical results, which show that our approaches can recover the fine-scale solution with a few degrees of freedom. The proposed methods can, in general, be used for more general nonlinear problems, where one needs nonlinear local spectral decomposition.

Citation: Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks & Heterogeneous Media, 2017, 12 (4) : 619-642. doi: 10.3934/nhm.2017025
References:
[1]

A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Mathematics of Computation, 83 (2014), 513-536. doi: 10.1090/S0025-5718-2013-02758-5.

[2]

G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, SIAM J. Multiscale Modeling and Simulation, 4 (2005), 790-812. doi: 10.1137/040611239.

[3]

G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518. doi: 10.1137/0523084.

[4]

T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal., 42 (2004), 576-598 (electronic). doi: 10.1137/S0036142902406636.

[5]

M. BarraultY. MadayN. C. Nguyen and A. T. Patera, An "empirical interpolation" method: Application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, 339 (2004), 667-672. doi: 10.1016/j.crma.2004.08.006.

[6]

Y. BazilevsV. M. CaloJ. A. CottrellT. J. R. HughesA. Reali and G. Scovazzi, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Computer Methods in Applied Mechanics and Engineering, 197 (2007), 173-201. doi: 10.1016/j.cma.2007.07.016.

[7]

L. BerlyandY. Gorb and A. Novikov, Discrete network approximation for highly-packed composites with irregular geometry in three dimensions, In Multiscale Methids in Science and Engineering, Spring, 44 (2005), 21-57. doi: 10.1007/3-540-26444-2_2.

[8]

L. Berlyand, A. G. Kolpakov and A. Novikov, Introduction to the Network Approximation Method for Materials Modeling Number 148. Cambridge University Press, 2013.

[9]

L. Berlyand and A. Novikov, Error of the network approximation for densely packed composites with irregular geometry, SIAM Journal on Mathematical Analysis, 34 (2002), 385-408. doi: 10.1137/S0036141001397144.

[10]

L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721. doi: 10.1007/s00205-010-0302-1.

[11]

V. M. CaloY. EfendievJ. Galvis and M. Ghommem, Multiscale empirical interpolation for solving nonlinear PDEs, Journal of Computational Physics, 278 (2014), 204-220. doi: 10.1016/j.jcp.2014.07.052.

[12]

V. M. CaloY. EfendievJ. Galvis and G. Li, Randomized oversampling for generalized multiscale finite element methods, Multiscale Model. Simul., 14 (2016), 482-501. doi: 10.1137/140988826.

[13]

V. ChiadóPiat and A. Defranceschi, Homogenization of monotone operators, Nonlinear Analysis: Theory, Methods & Applications, 14 (1990), 717-732. doi: 10.1016/0362-546X(90)90102-M.

[14]

C.-C. ChuI. G. Graham and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 79 (2010), 1915-1955. doi: 10.1090/S0025-5718-2010-02372-5.

[15]

E. ChungB. Cockburn and G. Fu, The staggered dg method is the limit of a hybridizable dg method, SIAM Journal on Numerical Analysis, 52 (2014), 915-932. doi: 10.1137/13091573X.

[16]

E. T. Chung, Y. Efendiev and W. T. Leung, An adaptive generalized multiscale discontinuous galerkin method (GMsDGM) for high-contrast flow problems arXiv preprint, arXiv:1409.3474,2014.

[17]

E. T. ChungY. Efendiev and W. T. Leung, Residual-driven online generalized multiscale finite element methods, J. Comput. Phys., 302 (2015), 176-190. doi: 10.1016/j.jcp.2015.07.068.

[18]

E. T. ChungY. Efendiev and G. Li, An adaptive GMsFEM for high-contrast flow problems, Journal of Computational Physics, 273 (2014), 54-76. doi: 10.1016/j.jcp.2014.05.007.

[19]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems volume 40. Siam, 2002.

[20]

B. CockburnJ. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems, SIAM Journal on Numerical Analysis, 47 (2009), 1319-1365. doi: 10.1137/070706616.

[21]

L. J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resour. Res., 27 (1991), 699-708. doi: 10.1029/91WR00107.

[22]

W. E and B. Engquist, Heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. doi: 10.4310/CMS.2003.v1.n1.a8.

[23]

Y. Efendiev and J. Galvis, Coarse-grid multiscale model reduction techniques for flows in heterogeneous media and applications, Numerical Analysis of Multiscale Problems, Lecture Notes in Computational Science and Engineering, 83 (2012), 97-125. doi: 10.1007/978-3-642-22061-6_4.

[24]

Y. EfendievJ. Galvis and T. Hou, Generalized multiscale finite element methods, Journal of Computational Physics, 251 (2013), 116-135. doi: 10.1016/j.jcp.2013.04.045.

[25]

Y. EfendievJ. GalvisS. Ki Kang and R. D. Lazarov, Robust multiscale iterative solvers for nonlinear flows in highly heterogeneous media, Numer. Math. Theory Methods Appl., 5 (2012), 359-383. doi: 10.4208/nmtma.2012.m1112.

[26]

Y. EfendievJ. GalvisG. Li and M. Presho, Generalized multiscale finite element methods. Oversampling strategies, International Journal for Multiscale Computational Engineering, accepted, 12 (2014), 465-484. doi: 10.1615/IntJMultCompEng.2014007646.

[27]

Y. EfendievJ. Galvis and X. H. Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions, Journal of Computational Physics, 230 (2011), 937-955. doi: 10.1016/j.jcp.2010.09.026.

[28]

Y. Efendiev and T. Hou, Multiscale Finite Element Methods: Theory and Applications Springer, 2009.

[29]

Y. EfendievT. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Comm. Math. Sci., 2 (2004), 553-589. doi: 10.4310/CMS.2004.v2.n4.a2.

[30]

Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations, SIAM J. Appl. Math., 65 (2004), 43-68. doi: 10.1137/S0036139903424886.

[31]

Y. EfendievJ. GalvisM Presho and J. Zhou, A multiscale enrichment procedure for nonlinear monotone operators, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 475-491. doi: 10.1051/m2an/2013116.

[32]

Y. EfendievR. LazarovM. Moon and K. Shi, A spectral multiscale hybridizable discontinuous Galerkin method for second order elliptic problems, Computer Methods in Applied Mechanics and Engineering, 292 (2015), 243-256. doi: 10.1016/j.cma.2014.09.036.

[33]

J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model. Simul., 8 (2010), 1461-1483. doi: 10.1137/090751190.

[34]

J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces, Multiscale Model. Simul., 8 (2010), 1621-1644. doi: 10.1137/100790112.

[35]

R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problémes de dirichlet non linéaires, Revue française d'automatique, informatique, recherche opérationnelle. Analyse numérique, 9 (1975), 41-76.

[36]

P. Henning, Heterogeneous multiscale finite element methods for advection-diffusion and nonlinear elliptic multiscale problems, Münster: Univ. Münster, Mathematisch-Naturwissenschaftliche Fakultät, Fachbereich Mathematik und Informatik (Diss. ). ii (2011), page 63.

[37]

P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems, Discrete and Continuous Dynamical Systems-Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 119-150. doi: 10.3934/dcdss.2015.8.119.

[38]

T. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Computer Methods in Applied Mechanics and Engineering, 127 (1995), 387-401. doi: 10.1016/0045-7825(95)00844-9.

[39]

T. J. R. HughesG. FeijooL. Mazzei and J. Quincy, The variational multiscale methoda paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3-24. doi: 10.1016/S0045-7825(98)00079-6.

[40]

T. J. R. Hughes and G. Sangalli, Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods, SIAM Journal on Numerical Analysis, 45 (2007), 539-557. doi: 10.1137/050645646.

[41]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals Springer-Verlag, Berlin, 1994.

[42]

J. L. LionsD. LukkassenL. E. Persson and P. Wall, Reiterated homogenization of nonlinear monotone operators, Chinese Annals of Mathematics, 22 (2001), 1-12. doi: 10.1142/S0252959901000024.

[43]

P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, Journal of the American Mathematical Society, 18 (2005), 121-156. doi: 10.1090/S0894-0347-04-00469-2.

[44]

G. Nguetseng and H. Nnang, Homogenization of nonlinear monotone operators beyond the periodic setting, Electr. J. of Diff. Eqns, 36 (2003), 1-24.

[45]

H. OwhadiL. Zhang and L. Berlyand, Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 517-552. doi: 10.1051/m2an/2013118.

[46]

A. A. Pankov, G-convergence and Homogenization of Nonlinear Partial Differential Operators volume 422. Mathematics and its Applications, 422. Kluwer Academic Publishers, Dordrecht, 1997.

[47]

G. Papanicolau, A. Bensoussan and J. -L. Lions, Asymptotic Analysis for Periodic Structures Elsevier, 1978.

[48]

M. Presho and S. Ye, Reduced-order multiscale modeling of nonlinear p-Laplacian flows in high-contrast media, Computational Geosciences, 19 (2015), 921-932. doi: 10.1007/s10596-015-9504-9.

[49]

X. H. WuY. Efendiev and T. Y. Hou, Analysis of upscaling absolute permeability, Discrete and Continuous Dynamical Systems, Series B., 2 (2002), 185-204. doi: 10.3934/dcdsb.2002.2.185.

[50]

E. Zeidler, Nonlinear Functional Analysis and Its Applications: Ⅲ: Variational Methods and Optimization Springer-Verlag, New York, 1985.

show all references

References:
[1]

A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Mathematics of Computation, 83 (2014), 513-536. doi: 10.1090/S0025-5718-2013-02758-5.

[2]

G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, SIAM J. Multiscale Modeling and Simulation, 4 (2005), 790-812. doi: 10.1137/040611239.

[3]

G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518. doi: 10.1137/0523084.

[4]

T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal., 42 (2004), 576-598 (electronic). doi: 10.1137/S0036142902406636.

[5]

M. BarraultY. MadayN. C. Nguyen and A. T. Patera, An "empirical interpolation" method: Application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, 339 (2004), 667-672. doi: 10.1016/j.crma.2004.08.006.

[6]

Y. BazilevsV. M. CaloJ. A. CottrellT. J. R. HughesA. Reali and G. Scovazzi, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Computer Methods in Applied Mechanics and Engineering, 197 (2007), 173-201. doi: 10.1016/j.cma.2007.07.016.

[7]

L. BerlyandY. Gorb and A. Novikov, Discrete network approximation for highly-packed composites with irregular geometry in three dimensions, In Multiscale Methids in Science and Engineering, Spring, 44 (2005), 21-57. doi: 10.1007/3-540-26444-2_2.

[8]

L. Berlyand, A. G. Kolpakov and A. Novikov, Introduction to the Network Approximation Method for Materials Modeling Number 148. Cambridge University Press, 2013.

[9]

L. Berlyand and A. Novikov, Error of the network approximation for densely packed composites with irregular geometry, SIAM Journal on Mathematical Analysis, 34 (2002), 385-408. doi: 10.1137/S0036141001397144.

[10]

L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for Rational Mechanics and Analysis, 198 (2010), 677-721. doi: 10.1007/s00205-010-0302-1.

[11]

V. M. CaloY. EfendievJ. Galvis and M. Ghommem, Multiscale empirical interpolation for solving nonlinear PDEs, Journal of Computational Physics, 278 (2014), 204-220. doi: 10.1016/j.jcp.2014.07.052.

[12]

V. M. CaloY. EfendievJ. Galvis and G. Li, Randomized oversampling for generalized multiscale finite element methods, Multiscale Model. Simul., 14 (2016), 482-501. doi: 10.1137/140988826.

[13]

V. ChiadóPiat and A. Defranceschi, Homogenization of monotone operators, Nonlinear Analysis: Theory, Methods & Applications, 14 (1990), 717-732. doi: 10.1016/0362-546X(90)90102-M.

[14]

C.-C. ChuI. G. Graham and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 79 (2010), 1915-1955. doi: 10.1090/S0025-5718-2010-02372-5.

[15]

E. ChungB. Cockburn and G. Fu, The staggered dg method is the limit of a hybridizable dg method, SIAM Journal on Numerical Analysis, 52 (2014), 915-932. doi: 10.1137/13091573X.

[16]

E. T. Chung, Y. Efendiev and W. T. Leung, An adaptive generalized multiscale discontinuous galerkin method (GMsDGM) for high-contrast flow problems arXiv preprint, arXiv:1409.3474,2014.

[17]

E. T. ChungY. Efendiev and W. T. Leung, Residual-driven online generalized multiscale finite element methods, J. Comput. Phys., 302 (2015), 176-190. doi: 10.1016/j.jcp.2015.07.068.

[18]

E. T. ChungY. Efendiev and G. Li, An adaptive GMsFEM for high-contrast flow problems, Journal of Computational Physics, 273 (2014), 54-76. doi: 10.1016/j.jcp.2014.05.007.

[19]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems volume 40. Siam, 2002.

[20]

B. CockburnJ. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems, SIAM Journal on Numerical Analysis, 47 (2009), 1319-1365. doi: 10.1137/070706616.

[21]

L. J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resour. Res., 27 (1991), 699-708. doi: 10.1029/91WR00107.

[22]

W. E and B. Engquist, Heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. doi: 10.4310/CMS.2003.v1.n1.a8.

[23]

Y. Efendiev and J. Galvis, Coarse-grid multiscale model reduction techniques for flows in heterogeneous media and applications, Numerical Analysis of Multiscale Problems, Lecture Notes in Computational Science and Engineering, 83 (2012), 97-125. doi: 10.1007/978-3-642-22061-6_4.

[24]

Y. EfendievJ. Galvis and T. Hou, Generalized multiscale finite element methods, Journal of Computational Physics, 251 (2013), 116-135. doi: 10.1016/j.jcp.2013.04.045.

[25]

Y. EfendievJ. GalvisS. Ki Kang and R. D. Lazarov, Robust multiscale iterative solvers for nonlinear flows in highly heterogeneous media, Numer. Math. Theory Methods Appl., 5 (2012), 359-383. doi: 10.4208/nmtma.2012.m1112.

[26]

Y. EfendievJ. GalvisG. Li and M. Presho, Generalized multiscale finite element methods. Oversampling strategies, International Journal for Multiscale Computational Engineering, accepted, 12 (2014), 465-484. doi: 10.1615/IntJMultCompEng.2014007646.

[27]

Y. EfendievJ. Galvis and X. H. Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions, Journal of Computational Physics, 230 (2011), 937-955. doi: 10.1016/j.jcp.2010.09.026.

[28]

Y. Efendiev and T. Hou, Multiscale Finite Element Methods: Theory and Applications Springer, 2009.

[29]

Y. EfendievT. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Comm. Math. Sci., 2 (2004), 553-589. doi: 10.4310/CMS.2004.v2.n4.a2.

[30]

Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations, SIAM J. Appl. Math., 65 (2004), 43-68. doi: 10.1137/S0036139903424886.

[31]

Y. EfendievJ. GalvisM Presho and J. Zhou, A multiscale enrichment procedure for nonlinear monotone operators, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 475-491. doi: 10.1051/m2an/2013116.

[32]

Y. EfendievR. LazarovM. Moon and K. Shi, A spectral multiscale hybridizable discontinuous Galerkin method for second order elliptic problems, Computer Methods in Applied Mechanics and Engineering, 292 (2015), 243-256. doi: 10.1016/j.cma.2014.09.036.

[33]

J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model. Simul., 8 (2010), 1461-1483. doi: 10.1137/090751190.

[34]

J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces, Multiscale Model. Simul., 8 (2010), 1621-1644. doi: 10.1137/100790112.

[35]

R. Glowinski and A. Marroco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problémes de dirichlet non linéaires, Revue française d'automatique, informatique, recherche opérationnelle. Analyse numérique, 9 (1975), 41-76.

[36]

P. Henning, Heterogeneous multiscale finite element methods for advection-diffusion and nonlinear elliptic multiscale problems, Münster: Univ. Münster, Mathematisch-Naturwissenschaftliche Fakultät, Fachbereich Mathematik und Informatik (Diss. ). ii (2011), page 63.

[37]

P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems, Discrete and Continuous Dynamical Systems-Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, 8 (2015), 119-150. doi: 10.3934/dcdss.2015.8.119.

[38]

T. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Computer Methods in Applied Mechanics and Engineering, 127 (1995), 387-401. doi: 10.1016/0045-7825(95)00844-9.

[39]

T. J. R. HughesG. FeijooL. Mazzei and J. Quincy, The variational multiscale methoda paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3-24. doi: 10.1016/S0045-7825(98)00079-6.

[40]

T. J. R. Hughes and G. Sangalli, Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods, SIAM Journal on Numerical Analysis, 45 (2007), 539-557. doi: 10.1137/050645646.

[41]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals Springer-Verlag, Berlin, 1994.

[42]

J. L. LionsD. LukkassenL. E. Persson and P. Wall, Reiterated homogenization of nonlinear monotone operators, Chinese Annals of Mathematics, 22 (2001), 1-12. doi: 10.1142/S0252959901000024.

[43]

P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, Journal of the American Mathematical Society, 18 (2005), 121-156. doi: 10.1090/S0894-0347-04-00469-2.

[44]

G. Nguetseng and H. Nnang, Homogenization of nonlinear monotone operators beyond the periodic setting, Electr. J. of Diff. Eqns, 36 (2003), 1-24.

[45]

H. OwhadiL. Zhang and L. Berlyand, Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 517-552. doi: 10.1051/m2an/2013118.

[46]

A. A. Pankov, G-convergence and Homogenization of Nonlinear Partial Differential Operators volume 422. Mathematics and its Applications, 422. Kluwer Academic Publishers, Dordrecht, 1997.

[47]

G. Papanicolau, A. Bensoussan and J. -L. Lions, Asymptotic Analysis for Periodic Structures Elsevier, 1978.

[48]

M. Presho and S. Ye, Reduced-order multiscale modeling of nonlinear p-Laplacian flows in high-contrast media, Computational Geosciences, 19 (2015), 921-932. doi: 10.1007/s10596-015-9504-9.

[49]

X. H. WuY. Efendiev and T. Y. Hou, Analysis of upscaling absolute permeability, Discrete and Continuous Dynamical Systems, Series B., 2 (2002), 185-204. doi: 10.3934/dcdsb.2002.2.185.

[50]

E. Zeidler, Nonlinear Functional Analysis and Its Applications: Ⅲ: Variational Methods and Optimization Springer-Verlag, New York, 1985.

Figure 1.  Illustration of a multiscale discretization.
Figure 2.  Illustration of a coarse neighborhood and elements.
Figure 3.  Illustration of the high-contrast permeability field $κ_1(x)$.
Figure 4.  Relative error vs $L_i$ for p=3, 4, 5, 6.
Figure 5.  FEM v.s. GMsFEM node-wise solutions, p=3, DOF=324.
Figure 6.  Illustration of the high-contrast permeability field $\kappa_3(x)$.
Table 1.  Relative errors for $p=3, 4, 5, 6$ using different numbers of cross basis.
$L_i$ (DOF)$p=3$
$L_p$ errorEnergy error
1(81)9.52 %41.03 %
2(162)6.45 %34.38 %
3(243)5.76 %27.76 %
4(324)0.52 %6.55 %
5(405)0.45 %5.15 %
$L_i$ (DOF)$p=4$
$L_p$ errorEnergy error
1(81)10.88 %42.35 %
2(162)6.47 %32.93 %
3(243)5.12 %24.13 %
4(324)0.92 %8.57 %
5(405)0.82 %6.65 %
$L_i$ (DOF)$p=5$
$L_p$ errorEnergy error
1(81)10.12 %40.46 %
2(162)7.71 %34.05 %
3(243)5.17 %27.88 %
4(324)0.94 %9.94 %
5(405)0.81 %7.92 %
$L_i$ (DOF)$p=3$
$L_p$ errorEnergy error
1(81)8.95 %39.68 %
2(162)6.94 %30.92 %
3(243)4.37 %23.85 %
4(324)1.07 %8.70 %
5(405)0.91 %7.08 %
$L_i$ (DOF)$p=3$
$L_p$ errorEnergy error
1(81)9.52 %41.03 %
2(162)6.45 %34.38 %
3(243)5.76 %27.76 %
4(324)0.52 %6.55 %
5(405)0.45 %5.15 %
$L_i$ (DOF)$p=4$
$L_p$ errorEnergy error
1(81)10.88 %42.35 %
2(162)6.47 %32.93 %
3(243)5.12 %24.13 %
4(324)0.92 %8.57 %
5(405)0.82 %6.65 %
$L_i$ (DOF)$p=5$
$L_p$ errorEnergy error
1(81)10.12 %40.46 %
2(162)7.71 %34.05 %
3(243)5.17 %27.88 %
4(324)0.94 %9.94 %
5(405)0.81 %7.92 %
$L_i$ (DOF)$p=3$
$L_p$ errorEnergy error
1(81)8.95 %39.68 %
2(162)6.94 %30.92 %
3(243)4.37 %23.85 %
4(324)1.07 %8.70 %
5(405)0.91 %7.08 %
Table 2.  Values of $\Lambda_{*}$ and $1/\Lambda_{*}$ when $p=3$.
$L_i$$\Lambda_{*}$$1/\Lambda_{*}$
18.86e-41.13e3
22.59e-33.86e2
34.46e-32.24e2
41.55e26.44e-3
54.01e22.50e-3
$L_i$$\Lambda_{*}$$1/\Lambda_{*}$
18.86e-41.13e3
22.59e-33.86e2
34.46e-32.24e2
41.55e26.44e-3
54.01e22.50e-3
Table 3.  Relative energy errors and values of $1/\Lambda_{*}$ using $\kappa_2(x)$, $p=3$.
$L_i$Energy error$1/\Lambda_{*}$
144.15 %1.42e3
236.44 %4.04e2
327.99 %2.35e2
46.77 %6.49e-3
55.30 %2.50e-3
$L_i$Energy error$1/\Lambda_{*}$
144.15 %1.42e3
236.44 %4.04e2
327.99 %2.35e2
46.77 %6.49e-3
55.30 %2.50e-3
Table 4.  Relative energy errors and values of $1/\Lambda_{*}$ using $\kappa_3(x)$, $p=3$.
$L_i$Energy error$1/\Lambda_{*}$
147.08 %1.85e1
227.68 %4.64e0
320.81 %2.68e0
44.33 %2.26e-3
52.69 %1.01e-3
$L_i$Energy error$1/\Lambda_{*}$
147.08 %1.85e1
227.68 %4.64e0
320.81 %2.68e0
44.33 %2.26e-3
52.69 %1.01e-3
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