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Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations

Published online by Cambridge University Press:  03 June 2015

Yalchin Efendiev*
Affiliation:
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
Juan Galvis*
Affiliation:
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá D.C., Colombia
Guanglian Li*
Affiliation:
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843, USA
Michael Presho*
Affiliation:
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843, USA
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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.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2014

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