2024.04.17 10AM, ASTC 615
4월17일 수요일 / 첨단관 615호 / 문의: 박은재 교수님 (ejpark@yonsei.ac.kr)
10:00~10:50
Prof. Dr. Lars Diening (Universität Bielefeld, Fakultät für Mathematik)
Kacanov Iteration
The p-Laplace equation is one of the model equations for non-linear problems. Due to its non-linearity it is quite challenging to approximate its solution numerically in particular in the degenerate/singular case. Standard methods like gradient descent or Newton's method have significant problems to approximate the solution. We present an iterative, linear method that allows to solve the p-Laplace equation efficiently both for small and large exponents.
10:50~11:20
Julian Rolfes (Universität Bielefeld, Fakultät für Mathematik)
Pointwise gradient estimate of the Ritz projection
Let Ω⊂R^n be a convex polytope (n≤3). The Ritz projection is the best approximation, in the W01,2-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, that the gradient at any point in Ω is controlled by the Hardy--Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of interest in the analysis of PDEs immediately follows. Among those are weighted spaces, Orlicz spaces and Lorentz spaces.