In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on 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 even for rough coefficients. To solve the corresponding system of algebraic equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
Mots clés : finite element method, a priori error estimate, convergence, multiscale method, non-linear, computational homogenization, upscaling
@article{M2AN_2014__48_5_1331_0,
author = {Henning, Patrick and M\r{a}lqvist, Axel and Peterseim, Daniel},
title = {A localized orthogonal decomposition method for semi-linear elliptic problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1331--1349},
publisher = {EDP-Sciences},
volume = {48},
number = {5},
year = {2014},
doi = {10.1051/m2an/2013141},
mrnumber = {3264356},
zbl = {1300.35011},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013141/}
}
TY - JOUR AU - Henning, Patrick AU - Målqvist, Axel AU - Peterseim, Daniel TI - A localized orthogonal decomposition method for semi-linear elliptic problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 1331 EP - 1349 VL - 48 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013141/ DO - 10.1051/m2an/2013141 LA - en ID - M2AN_2014__48_5_1331_0 ER -
%0 Journal Article %A Henning, Patrick %A Målqvist, Axel %A Peterseim, Daniel %T A localized orthogonal decomposition method for semi-linear elliptic problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1331-1349 %V 48 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013141/ %R 10.1051/m2an/2013141 %G en %F M2AN_2014__48_5_1331_0
Henning, Patrick; Målqvist, Axel; Peterseim, Daniel. A localized orthogonal decomposition method for semi-linear elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1331-1349. doi : 10.1051/m2an/2013141. http://www.numdam.org/articles/10.1051/m2an/2013141/
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