# IRTG Bielefeld-Seoul Online Seminar

The seminar takes place on selected Mondays 9-10 am German time (4-5 pm Korean time, from November on: 5-6 pm Korean time).

Zoom: https://uni-bielefeld.zoom.us/j/96823879152?pwd=VmRmSUtQMUpYdE44U1UwdmxwSTB5dz09
ID: 968 2387 9152
Kenncode: 2235

## October 4, 2021

Tabea Tscherpel (Bielefeld): On the Sobolev stability of the $L^2$-Projection mapping to finite element spaces

Abstract: The $L^2$-projection mapping to Lagrange finite element spaces is a crucial tool in numerical analysis. Especially when dealing with evolution problems Sobolev stability of the $L^2$-projection is needed. While the proof of Sobolev stability is straightforward for uniform meshes, it is much more challenging for graded meshes. We tackle this difficulty by investigating the decay properties of the $L^2$-projection analytically. Inspired by the miracle of extrapolation we introduce a maximal operator. This allows us to obtain equivalence of decay and certain weighted estimates, that eventually lead to Sobolev stability. We present recent results in 2D and 3D for a wide range of polynomial degrees and graded meshes. Those include a large class of meshes used in practical applications.

This is joint work with Lars Diening and Johannes Storn from Bielefeld University (Germany).

## September 6, 2021

Minhyun Kim (Bielefeld): Local regularity theory for nonlocal equations

Abstract: We introduce several nonlocal equations which correspond to classical local models, attempting to develop parallel or unified theory. We study local regularity properties such as local boundedness, weak Harnack inequality, and local Hölder regularity of weak solutions to these nonlocal equations.