A localized orthogonal decomposition method for semi-linear elliptic problems∗∗∗. (13th August 2014)

• Record Type: Journal Article
• Authors: Henning, Patrick; Målqvist, Axel; Peterseim, Daniel
• Is Part Of: Mathematical modelling and numerical analysis. Volume 48:Part 5(2014)
• Page Start: 1331
• Page End: 1349
• Publication Date: 2014-08-13
• Subjects: Finite element method, -- a priori error estimate, -- convergence, -- multiscale method, -- non-linear, -- computational homogenization, -- upscaling; Numerical analysis -- Periodicals; Mathematical models -- Periodicals; 510
• DOI: 10.1051/m2an/2013141
• Languages: English
• ISSNs: 0764-583X

Abstract

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 H 1 -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.