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

Eric Chung; Yalchin Efendiev; Ke Shi; Shuai Ye; Eric Chung; Yalchin Efendiev; Ke Shi; Shuai Ye

doi:10.3934/nhm.2017025

Networks and Heterogeneous Media

2017, Volume 12, Issue 4: 619-642. doi: 10.3934/nhm.2017025

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

Eric Chung ^{1
,},
Yalchin Efendiev ^{2
,},
Ke Shi ^{3
,},
Shuai Ye ^{2
,}

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: 01 January 2016 Revised: 01 October 2016
Primary: 65N99

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.
- Generalized Multiscale Finite Element method,
- multiscale,
- nonlinear monotone elliptic problem,
- $p$-Laplacian,
- high-contrast
Citation: Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media[J]. Networks and Heterogeneous Media, 2017, 12(4): 619-642. doi: 10.3934/nhm.2017025

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Abstract

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.

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