A localized orthogonal decomposition method for semi-linear elliptic problems∗∗∗. (13th August 2014) - Henning, Patrick; Målqvist, Axel; Peterseim, Daniel
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.