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

Published online by Cambridge University Press:  13 August 2014

Patrick Henning ,
Axel Målqvist  and
Daniel Peterseim
Patrick Henning
Affiliation:
Department of Information Technology, Uppsala University, Box 337, 75105 Uppsala, Sweden.
Axel Målqvist
Affiliation:
Department of Information Technology, Uppsala University, Box 337, 75105 Uppsala, Sweden.
Daniel Peterseim
Affiliation:
Institut für Numerische Simulation der Universität Bonn, Wegelerstr. 66, 53123 Bonn, Germany.. patrick.henning@uni-muenster.de
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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.

Keywords

Finite element methoda priori error estimateconvergencemultiscale methodnon-linearcomputational homogenizationupscaling
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1331 - 1349
Copyright
© EDP Sciences, SMAI 2014

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