A rigorous Multiscale Method for semi-linear elliptic problems
Patrick Henning, Axel Målqvist, Daniel Peterseim · arXiv (Cornell University) · 2012
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problemswith heterogeneous andhighly variable coefficient functions. For this purposewe construct a generalized finiteelement basisthat spansalow dimensionalmultiscale space. The basis is assembled by performing localized linear fine-scale computations in small patches that have a diameter of order Hlog(H −1) 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. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.