Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Numerical homogenization for nonlinear multiscale problems

Talk by Dr. Barbara Verfürth

CANCELLED**Date:** 24.11.21 **Time:** 16.15 - 17.15 **Room:** ETH HG E 1.2

Many applications, such as geophysical flow problems or scattering from Kerr-type media, require the combination of nonlinear material laws and multiscale features, which together pose a huge computational challenge. In this talk, we discuss how to construct a problem-adapted multiscale basis in a linearized and localized fashion for nonlinear problems such as the quasilinear diffusion equation or the nonlinear Helmholtz equation. For this, we will adapt two different perspectives: (a) determining a fixed multiscale space for the nonlinear problem or (b) adaptively and iteratively updating the multiscale space during an iteration scheme for the nonlinear problem. We prove optimal error estimates for the corresponding generalized finite element methods. 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. Numerical examples show very promising results illustrating the theoretical convergence rates.