Numerical methods for multiscale problems with coefficient changes (Barbara Verfürth, University of Bonn)

Problems with multiple spatial scales occur in many applications, for instance when considering modern composite materials. Numerically, such problems are challenging as standard discretization would lead to huge systems. Instead, computational multiscale methods construct their basis functions from local solutions of the PDE, thereby generating problem-adapted approximation spaces. While this yields good approximations already on coarse meshes and is very efficient if many right-hand sides need to be studied, the problem-adapted approach becomes more difficult when the multiscale coefficients change themselves. In this talk, we consider the  Localized Decomposition Method (LOD) and discuss some recent ideas how to reduce this burden. As exemplary situation we concentrate on  coefficients with random defects where we present a so-called offline-online strategy. We present a priori error estimates as well as illustrative numerical experiments.