1 Department of Information Technology, Uppsala University, Box
337, 75105 Uppsala, Sweden.
2 Institut für Numerische Simulation der Universität Bonn, Wegelerstr. 66, 53123 Bonn, Germany.
Revised: 11 June 2013
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.
Mathematics Subject Classification: 35J15 / 65N12 / 65N30
Key words: Finite element method / a priori error estimate / convergence / multiscale method / non-linear / computational homogenization / upscaling
© EDP Sciences, SMAI 2014