Naslov
Saturation assumptions for Rayleigh--Ritz eigenvalue approximations

Autori
Grubišić, Luka

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Skup
Fourth Conference on Applied Mathematics and Scientific Computing

Mjesto i datum
Brijuni, Hrvatska, 19.06.2005. - 24.06.2005

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
finite element method; eigenvalue estimates; saturation assumption; adaptive mesh refinement

Sažetak
We are primarily concerned with an analysis of finite element methods for the eigenvalue/eigenvector problem for a selfadjoint elliptic operator. A saturation assumption expresses and quantifies, through a saturation constant, the desired quality in any approximation method: Enlarged test space leads to better approximations. In particular, one defines --- with a help of a saturation assumption --- discrete \textit{; ; ; a posteriori}; ; ; error estimates for elliptic boundary value problems which are not $H^2$ regular. This type of analysis is a particularly important step on a way towards an adaptive mesh refinement procedure. Only recently have Doerfler and Nochetto revealed a structure of such a saturation constant for a case of a boundary value problem. We adapt and apply the analysis of Doerfler and Nochetto to an analysis of the eigenvalue problem by the means of the Ritz-vector residuum. We also derive a class of Temple--Kato eigenvalue estimates. The eigenvalue estimates are accompanied by a $\sin\Theta$-like result for the accompanying eigenvectors. Our new residuum-based saturation constant will be compared with the saturation constant, featured in the Neymeyr's analysis of the Rayleigh-Ritz eigenvalue approximations. It will be shown that our discrete residuum estimate represents a first order estimate of the complete Ritz-vector residuum. This strongly corroborates the experimental results which were reported by Neymeyr. At the end of the lecture we will present some numerical results to illustrate the developed theory.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA