A Newton-scheme framework for multiscale methods for nonlinear elliptic homogenization problems

Abstract

In this contribution, we present a very general framework for formulating multiscale methods for nonlinear elliptic homogenization problems. The framework is based on a very general coupling of one macroscopic equation with several localized ne-scale problems. In particular, we recover the Heterogeneous Multiscale Method (HMM), the Multiscale Finite Element Method (MsFEM) and the Variational Multiscale Method (VMM) from the framework. In order to solve the arising equations, we also present a solution algorithm that is based on a Newton scheme with damping.