A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media Academic Article uri icon

abstract

  • American Institute of Mathematical Sciences. 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 sum- marized 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 snap- shot 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.

published proceedings

  • Networks & Heterogeneous Media

author list (cited authors)

  • Chung, E., Efendiev, Y., Shi, K. e., & Ye, S.

publication date

  • January 1, 2017 11:11 AM