# American Institute of Mathematical Sciences

December  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
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##### References:
Illustration of a multiscale discretization.
Illustration of a coarse neighborhood and elements.
Illustration of the high-contrast permeability field $κ_1(x)$.
Relative error vs $L_i$ for p=3, 4, 5, 6.
FEM v.s. GMsFEM node-wise solutions, p=3, DOF=324.
Illustration of the high-contrast permeability field $\kappa_3(x)$.
Relative errors for $p=3, 4, 5, 6$ using different numbers of cross basis.
 $L_i$ (DOF) $p=3$ $L_p$ error Energy 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) $p=4$ $L_p$ error Energy 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) $p=5$ $L_p$ error Energy 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) $p=3$ $L_p$ error Energy 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) $p=3$ $L_p$ error Energy 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) $p=4$ $L_p$ error Energy 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) $p=5$ $L_p$ error Energy 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) $p=3$ $L_p$ error Energy 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 %
Values of $\Lambda_{*}$ and $1/\Lambda_{*}$ when $p=3$.
 $L_i$ $\Lambda_{*}$ $1/\Lambda_{*}$ 1 8.86e-4 1.13e3 2 2.59e-3 3.86e2 3 4.46e-3 2.24e2 4 1.55e2 6.44e-3 5 4.01e2 2.50e-3
 $L_i$ $\Lambda_{*}$ $1/\Lambda_{*}$ 1 8.86e-4 1.13e3 2 2.59e-3 3.86e2 3 4.46e-3 2.24e2 4 1.55e2 6.44e-3 5 4.01e2 2.50e-3
Relative energy errors and values of $1/\Lambda_{*}$ using $\kappa_2(x)$, $p=3$.
 $L_i$ Energy error $1/\Lambda_{*}$ 1 44.15 % 1.42e3 2 36.44 % 4.04e2 3 27.99 % 2.35e2 4 6.77 % 6.49e-3 5 5.30 % 2.50e-3
 $L_i$ Energy error $1/\Lambda_{*}$ 1 44.15 % 1.42e3 2 36.44 % 4.04e2 3 27.99 % 2.35e2 4 6.77 % 6.49e-3 5 5.30 % 2.50e-3
Relative energy errors and values of $1/\Lambda_{*}$ using $\kappa_3(x)$, $p=3$.
 $L_i$ Energy error $1/\Lambda_{*}$ 1 47.08 % 1.85e1 2 27.68 % 4.64e0 3 20.81 % 2.68e0 4 4.33 % 2.26e-3 5 2.69 % 1.01e-3
 $L_i$ Energy error $1/\Lambda_{*}$ 1 47.08 % 1.85e1 2 27.68 % 4.64e0 3 20.81 % 2.68e0 4 4.33 % 2.26e-3 5 2.69 % 1.01e-3
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