A localized orthogonal decomposition method for semi-linear elliptic problems

TL;DR

This paper introduces a multiscale finite element method for semi-linear elliptic problems with complex coefficients, providing rigorous convergence analysis and an efficient Newton-based solver.

Contribution

It develops a localized orthogonal decomposition approach with a new multiscale basis for semi-linear elliptic problems, ensuring linear convergence without assumptions on coefficient oscillations.

Findings

Linear convergence of the H1-error with respect to coarse mesh size

Efficient Newton scheme for solving multiscale equations

Applicable to problems with highly variable coefficients

Abstract

In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations in 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. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.