Kolloquiumsvortrag: Prof. Dr. Jens Markus Melenk, TU Wien / 12.02.16

12.02.2016 von 14:15 bis 15:45

Institut für Informatik, Ludewig-Meyn-Str. 2, 24118 Kiel, Übungsraum 2

Titel: Stability and convergence of Galerkin discretizations of the Helmholtz equation


We consider boundary value problems for the Helmholtz equation at large wave numbers $k$. In order to understand how the wave number $k$ affects the convergence properties of discretizations of such problems, we develop a regularity theory for the Helmholtz equation that is explicit in $k$. At the heart of our analysis is the decomposition of solutions into two components: the first component is an analytic, but highly oscillatory function and the second one has finite regularity but features wavenumber-independent bounds. This new understanding of the solution structure opens the door to the analysis of discretizations of the Helmholtz equation that are explicit in their dependence on the wavenumber $k$. As a first example, we show for a conforming high order finite element method that quasi-optimality is guaranteed if (a) the approximation order $p$ is selected as $p = O(\log k)$ and (b) the mesh size $h$ is such that $kh/p$ is small. The work is joint with Stefan Sauter (Zurich).

Prof. Steffen Börm

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