NDLI: $hp$-Finite Elements for Elliptic Eigenvalue Problems: Error Estimates Which Are Explicit with Respect to $\lambda$, h, and p

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Sauter, S.
Copyright Year	2010
Abstract	Convergence rates for finite element discretizations of elliptic eigenvalue problems in the literature usually are of the following form: If the mesh width h is fine enough, then the eigenvalues, resp., eigenfunctions, converge at some well-defined rate. In this paper, we will determine the maximal mesh width $h_{0}$more precisely the minimal dimension of a finite element spaceso that the asymptotic convergence estimates hold for $h\leq h_{0}$. This mesh width will depend on the size and spacing of the exact eigenvalues, the spatial dimension, and the local polynomial degree of the finite element space. For example, in the one-dimensional case, the condition $\lambda^{3/4}h_{0}\lesssim1$ is sufficient for piecewise linear finite elements to compute an eigenvalue $\lambda$ with optimal convergence rates as $h_{0}\geq h\rightarrow0$. It will turn out that the condition for eigenfunctions is slightly more restrictive. Furthermore, we will analyze the dependence of the ratio of the errors of the Galerkin approximation and of the best approximation of an eigenfunction on $\lambda$ and h. In this paper, the error estimates for the eigenvalue/-function are limited to the selfadjoint case. However, the regularity theory and approximation property cover also the nonselfadjoint case and, hence, pave the way towards the error analysis of nonselfadjoint eigenvalue/-function problems.
Starting Page	95
Ending Page	108
Page Count	14
File Format	PDF
ISSN	00361429
DOI	10.1137/070702515
e-ISSN	10957170
Issue Number	1
Volume Number	48
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-04-02
Access Restriction	Subscribed
Subject Keyword	Stability and convergence of numerical methods elliptic eigenvalue problems Eigenvalue problems convergence rates Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods finite elements
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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