A Newton-Scheme Framework for Multiscale Methods for Nonlinear Elliptic Problems

Patrick Henning
Institut für Numerische und Angewandte Mathematik
Mathematik und Informatik der Universität Münster
University of Münster
Münster, Germany


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 fine-scale problems. In particular, we recover existing methods like the Heterogeneous Multiscale Method (HMM) and the Multiscale Finite Element Method (MsFEM) 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.