Commun. Comput. Phys., 15 (2014), pp. 733-755.


Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations

Yalchin Efendiev 1, Juan Galvis 2, Guanglian Li 3, Michael Presho 3*

1 Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia; Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843, USA.
2 Departamento de Matematicas, Universidad Nacional de Colombia, Bogota D.C., Colombia.
3 Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843, USA.

Received 2 March 2013; Accepted (in revised version) 4 October 2013
Available online 3 December 2013
doi:10.4208/cicp.020313.041013a

Abstract

In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.

AMS subject classifications: 35J60, 65N30

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Key words: Generalized multiscale finite element method, nonlinear equations, high-contrast.

*Corresponding author.
Email: efendiev@math.tamu.edu (Y. Efendiev), jcgalvisa@unal.edu.co (J. Galvis), lotusli@math.tamu.edu (G. Li), mpresho@math.tamu.edu (M. Presho)
 

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