Barbara Verfürth: Numerical homogenization of nonlinear multiscale diffusion problems

Abstract:

Many applications, such as geophysical flow problems, require the combination of nonlinear material laws and multiscale features, which together pose a huge computational challenge.
In this talk, we present a simple yet effective approach on how to construct a problem-adapted multiscale basis in a linearized and localized fashion for strongly monotone quasilinear problems.
The corresponding generalized finite element method -- in Galerkin as well as Petrov-Galerkin formulation -- gives optimal error estimates up to linearization errors. In particular, neither higher regularity of the exact solution nor structural properties of the coefficients such as scale separation or periodicity need to be assumed. Some ideas on an adaptive strategy when to compute the problem-adapted basis with a new linearization will also be discussed. Numerical examples show very promising results confirming the theoretical convergence rates and showing the possibility to generalize beyond monotone problems.