| 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. |
