A localized orthogonal decomposition method for semi-linear elliptic problems

Patrick Henning; Axel Målqvist; Daniel Peterseim

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

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

Abstract

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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.

How to cite

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Henning, 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 -

References

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