Geodesic
A localized orthogonal decomposition method for semi-linear elliptic problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1331-1349

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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.

DOI : 10.1051/m2an/2013141
Classification : 35J15, 65N12, 65N30
Keywords: finite element method, a priori error estimate, convergence, multiscale method, non-linear, computational homogenization, upscaling 
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     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},
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     year = {2014},
     doi = {10.1051/m2an/2013141},
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     zbl = {1300.35011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013141/}
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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

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