In this paper we propose and analyze a localized orthogonal decomposition (LOD) methodfor solving semi-linear elliptic problems with heterogeneous and highly variablecoefficient functions. This Galerkin-type method is based on a generalized finite elementbasis that spans a low dimensional multiscale space. The basis is assembled by performinglocalized linear fine-scale computations on small patches that have a diameter of orderH | log (H)| where H is the coarse mesh size. Without any assumptions onthe type of the oscillations in the coefficients, we give a rigorous proof for a linearconvergence of the H1-error with respect to the coarse meshsize even for rough coefficients. To solve the corresponding system of algebraicequations, we propose an algorithm that is based on a damped Newton scheme in themultiscale space.
