A LOCALIZED ORTHOGONAL DECOMPOSITION METHOD FOR SEMI-LINEAR ELLIPTIC PROBLEMS

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 II vertical bar log(H)vertical bar 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.