Paper Info
Title
A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media.
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.
Year
DOI
Venue
2017
10.3934/nhm.2017025
NETWORKS AND HETEROGENEOUS MEDIA
Keywords
Field
DocType
Generalized Multiscale Finite Element method,multiscale,nonlinear monotone elliptic problem,p-Laplacian,high-contrast
Convergence (routing),Mathematical optimization,Nonlinear system,Mathematical analysis,Homogenization (chemistry),Matrix decomposition,Finite element method,Eigenvalues and eigenvectors,Mathematics,Monotone polygon,p-Laplacian
Journal
Volume
Issue
ISSN
12
4
1556-1801
Citations 
PageRank 
References 
2
0.47
16
Authors
4
Name
Order
Citations
PageRank
Eric T. Chung138846.61
Yalchin Efendiev258167.04
Ke Shi31047.03
Shuai Ye430.83