# A localized orthogonal decomposition method for semi-linear elliptic problems

Patrick Henning; Axel Målqvist; Daniel Peterseim

- Volume: 48, Issue: 5, page 1331-1349
- ISSN: 0764-583X

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topHenning, Patrick, Målqvist, Axel, and Peterseim, Daniel. "A localized orthogonal decomposition method for semi-linear elliptic problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.5 (2014): 1331-1349. <http://eudml.org/doc/273096>.

@article{Henning2014,

abstract = {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.},

author = {Henning, Patrick, Målqvist, Axel, Peterseim, Daniel},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {finite element method; a priori error estimate; convergence; multiscale method; non-linear; computational homogenization; upscaling},

language = {eng},

number = {5},

pages = {1331-1349},

publisher = {EDP-Sciences},

title = {A localized orthogonal decomposition method for semi-linear elliptic problems},

url = {http://eudml.org/doc/273096},

volume = {48},

year = {2014},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 5

SP - 1331

EP - 1349

AB - 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.

LA - eng

KW - finite element method; a priori error estimate; convergence; multiscale method; non-linear; computational homogenization; upscaling

UR - http://eudml.org/doc/273096

ER -

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