Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations

TL;DR

This paper extends the GMsFEM framework to nonlinear elliptic equations with high-contrast coefficients, employing linearization and constructing coarse solution spaces for CG and DG methods, demonstrating effective error control through numerical examples.

Contribution

It introduces a systematic approach to applying GMsFEM to nonlinear elliptic equations using linearization and coarse space construction for CG and DG formulations.

Findings

Both CG and DG methods show predictable error decline with increasing coarse space dimension.

Numerical examples confirm the effectiveness of the proposed methods.

The approach handles high-contrast coefficients efficiently.

Abstract

In this paper we use the GeneralizedMultiscale Finite ElementMethod (GMsFEM) framework, introduced in [20], 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.