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.