Talk given by Barbara Verfürth on Numerics for multiscale coefficients with random defects
Many modern materials exhibit multiscale structures. Imperfections in the production process can lead to material defects, which we model by certain random perturbations. In this talk, we are interested in numerically simulating the solutions to diffusion problems for any given sample of such a coefficient. First, we briefly discuss the effects of single defects on the solutions. Then, we propose a computational multiscale methods based upon problem-adapted spaces for the problem. To reduce the complexity of calculating a multiscale basis for each sample, we present an offline-online strategy and discuss its error analysis. The key idea is to pre-compute local stiffness matrices for a cleverly chosen set of coefficients offline which allows to assemble the system matrix for any sample quickly in the online phase. In the end, we discuss some further developments and connections. This is based on joint work with D. Kolombage, A. Målqvist, and A. Wayoff.