A localized orthogonal decomposition method for semi-linear elliptic problems
Numerical Analysis
2019-02-20 v2
Abstract
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations in small patches that have a diameter of order H |log H| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
Cite
@article{arxiv.1211.3551,
title = {A localized orthogonal decomposition method for semi-linear elliptic problems},
author = {Patrick Henning and Axel Malqvist and Daniel Peterseim},
journal= {arXiv preprint arXiv:1211.3551},
year = {2019}
}