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# Seminar Interpolation spaces - Detailseite

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Veranstaltungsart Veranstaltungsnummer Seminar 3314407 SoSe 2020 2 Freigegeben für Vorlesungsverzeichnis  Freigegeben englisch Digital

### Termine

Gruppe 1
Tag Zeit Rhythmus Dauer Raum Raum-
plan
Lehrperson Status Bemerkung fällt aus am Max. Teilnehmer
Mo. 11:00 bis 13:00 wöch Johann von Neumann-Haus - 1.115 Rudower Chaussee 25 (RUD25) - (Hör-/Lehrsäle ansteigend ohne Experimentierbühne) Carstensen findet statt     30
Gruppe 1:
Zur Zeit keine Belegung möglich

Zugeordnete Person
Zugeordnete Person Zuständigkeit
Carstensen, Carsten , Prof. verantwortlich
Studiengänge
Abschluss Studiengang LP Semester
Bachelor of Science  Mathematik Monobachelor ( POVersion: 2009 )   -
Bachelor of Science  Mathematik Monobachelor ( Vertiefung: kein LA; POVersion: 2014 )   -
Zuordnung zu Einrichtungen
Einrichtung
Mathematisch-Naturwissenschaftliche Fakultät, Institut für Mathematik
Kommentar Inhalt: The word interpolation is used for two different areas of mathematics. There is the elementary interpolation of polynomial or finite element spaces in numerical analysis and there is the interpolation of Hilbert (or Banach) spaces. The latter is the topic of this seminar because it is usually not taught in the Berlin curriculum but has important applications to the precise rate of convergence and the precise regularity of a solution of a partial differential equation. The topics of the seminar try to cover three aspects. (a) The definition of the real and complex interpolation of Hilbert spaces e.g. after the book of Luc Tartar on an introduction  to Sobolev spaces and interpolation spaces, Springer 2007. A related introduction is also given in an appendix of the standard Springer finite element book due to Brenner-Scott. The definitions are quite abstract but we like to work in this concept to mention at least regularity in (b): The solution of an elliptic  partial differential equation with smooth right-hand side in a polyhedral bounded Lipschitz domain$\Omega\subset \R^n$  belongs to some $H^{1+s} (\Omega)$ for which real s and white which definition of $H^s(\Omega)$? For example the function $r^\alpha$ in polar coordinates with exponent $\alpha$ belongs to some $H^s(\Omega)$, but what is the relation of $s,\alpha,n$? We will not prove but mention the decomposition theorem for the singular functions of the Laplacian, but apply it to deduce $\nabla u\in H^{s}$ for some $0‹s‹1$. The last aspect (c) revisits the finite element approximation with the $L2$ projection $\Pi_0$ onto piecewise constants. The Poincare inequality shows $\| \nabla u - \Pi_0\nabla u\|_{L2(\Omega)} \le h_\max/\pi | u |_{H^{2}(\Omega)}$ for an underlying partition into convex domains with maximal diameter $h_\max$ and $s=1$. The Pythagoras theorem leads to $\| \nabla u - \Pi_0\nabla u\|_{L2(\Omega)} \le | u |_{H^{1}(\Omega)}$ for $s=0$. So why does it follow from these relative elementary  observations that $\| \nabla u - \Pi_0\nabla u\|_{L2(\Omega)} \le (h_\max/\pi)^s | u |_{H^{1+s}(\Omega)}$ for $0\le s \le 1$? Combined with he precise regularity this leads to a non-integer convergence rate $s$ observed in the standard finite element analysis of conforming and nonconforming methods. The seminar is kept at a most elementary level to foster the understanding of the definition of $H^s(\Omega)$ for real $s$ by interpolation of Sobolev spaces.  The topics (a)-(c) form  is a fundamental aspect, typically put aside but relevant for higher lectures in partial differential equations and computational partial differential equations.

## Strukturbaum

Keine Einordnung ins Vorlesungsverzeichnis vorhanden. Veranstaltung ist aus dem Semester SoSe 2020. Aktuelles Semester: WiSe 2020/21.
Humboldt-Universität zu Berlin | Unter den Linden 6 | D-10099 Berlin