engleski

Saturation assumptions for Rayleigh--Ritz eigenvalue approximations

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.

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

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