Networks and Heterogeneous Media (NHM)

A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media
Pages: 619 - 642, Issue 4, December 2017

doi:10.3934/nhm.2017025      Abstract        References        Full text (547.2K)           Related Articles

Eric Chung - Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, SAR, China (email)
Yalchin Efendiev - Department of Mathematics, Texas A & M University, College Station, TX 77843, United States (email)
Ke Shi - Department of Mathematics & Statistics, Old Dominion University, Norfolk, VA 23529, United States (email)
Shuai Ye - Department of Mathematics, Texas A & M University, College Station, TX 77843, United States (email)

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.       
2 G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, SIAM J. Multiscale Modeling and Simulation, 4 (2005), 790-812.       
3 G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.       
4 T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal., 42 (2004), 576-598 (electronic).       
5 M. Barrault, Y. Maday, N. 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.       
6 Y. Bazilevs, V. M. Calo, J. A. Cottrell, T. J. R. Hughes, A. 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.       
7 L. Berlyand, Y. Gorb and A. Novikov, Discrete network approximation for highly-packed composites with irregular geometry in three dimensions, In Multiscale Methods in Science and Engineering, Springer, 44 (2005), 21-57.       
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.       
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.       
11 V. M. Calo, Y. Efendiev, J. Galvis and M. Ghommem, Multiscale empirical interpolation for solving nonlinear PDEs, Journal of Computational Physics, 278 (2014), 204-220.       
12 V. M. Calo, Y. Efendiev, J. Galvis and G. Li, Randomized oversampling for generalized multiscale finite element methods, Multiscale Model. Simul., 14 (2016), 482-501, http://arxiv.org/pdf/1409.7114.pdf.       
13 V. ChiadòPiat and A. Defranceschi, Homogenization of monotone operators, Nonlinear Analysis: Theory, Methods & Applications, 14 (1990), 717-732.       
14 C.-C. Chu, I. G. Graham and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 79 (2010), 1915-1955.       
15 E. Chung, B. 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.       
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. Chung, Y. Efendiev and W. T. Leung, Residual-driven online generalized multiscale finite element methods, J. Comput. Phys., 302 (2015), 176-190.       
18 E. T. Chung, Y. Efendiev and G. Li, An adaptive GMsFEM for high-contrast flow problems, Journal of Computational Physics, 273 (2014), 54-76.       
19 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, volume 40. Siam, 2002.       
20 B. Cockburn, J. 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.       
21 L. J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resour. Res., 27 (1991), 699-708.
22 W. E and B. Engquist, Heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132.       
23 Y. Efendiev and J. Galvis, Coarse-grid multiscale model reduction techniques for flows in heterogeneous media and applications, Chapter of Numerical Analysis of Multiscale Problems, Lecture Notes in Computational Science and Engineering, 83 (2012), 97-125.       
24 Y. Efendiev, J. Galvis and T. Hou, Generalized multiscale finite element methods, Journal of Computational Physics, 251 (2013), 116-135.       
25 Y. Efendiev, J. Galvis, S. 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.       
26 Y. Efendiev, J. Galvis, G. Li and M. Presho, Generalized multiscale finite element methods. Oversampling strategies, International Journal for Multiscale Computational Engineering, accepted, 12 (2014), 465-484.
27 Y. Efendiev, J. 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.       
28 Y. Efendiev and T. Hou, Multiscale Finite Element Methods: Theory and Applications, Springer, 2009.       
29 Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Comm. Math. Sci., 2 (2004), 553-589.       
30 Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations, SIAM J. Appl. Math., 65 (2004), 43-68.       
31 Y. Efendiev, J. Galvis, M Presho and J. Zhou, A multiscale enrichment procedure for nonlinear monotone operators, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 475-491.       
32 Y. Efendiev, R. Lazarov, M. 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.       
33 J. Galvis and Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model. Simul., 8 (2010), 1461-1483.       
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.       
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 problèmes 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.       
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.       
39 T. J. R. Hughes, G. Feijoo, L. Mazzei and J. Quincy, The variational multiscale method-a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3-24.       
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.       
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. Lions, D. Lukkassen, L. E. Persson and P. Wall, Reiterated homogenization of nonlinear monotone operators, Chinese Annals of Mathematics, 22 (2001), 1-12.       
43 P. Ming and P. Zhang, et al., Analysis of the heterogeneous multiscale method for elliptic homogenization problems, Journal of the American Mathematical Society, 18 (2005), 121-156.       
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. Owhadi, L. Zhang and L. Berlyand, Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 517-552.       
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.       
49 X. H. Wu, Y. Efendiev and T. Y. Hou, Analysis of upscaling absolute permeability, Discrete and Continuous Dynamical Systems, Series B., 2 (2002), 185-204.       
50 E. Zeidler, Nonlinear Functional Analysis and Its Applications: III: Variational Methods and Optimization, Springer-Verlag, New York, 1985.       

Go to top