A numerical existence proof of nodal lines for the first eigenfunction of the plate equation
type: Item
instance of: scholarly article
title: A numerical existence proof of nodal lines for the first eigenfunction of the plate equation
zbMATH Open document ID: 0854.65092
DOI: 10.1007/BF01781561
author: Christian Wieners
published in: Archiv der Mathematik
publication date: 1997-01-27T00:00:00Z
review text: We explain a numerical procedure to compute error bounds in \(H^2_0(\Omega)\) for eigenfunctions of elliptic eigenvalue problems of fourth order. Therefore, we compute a finite element approximation and an upper bound for the defect in \(H^{-2} (\Omega)\). Then, a theorem of Kato, eigenvalue inclusions and explicit embedding constants yield a pointwise error bound for the approximation. In order to control rounding errors, we use interval arithmetic. As an application, we prove the existence of nodal lines for the first eigenfunction of the clamped plate and for the buckling plate in a square.
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 74S05
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 74K20
zbMATH DE Number: 938555
zbMATH Keywords: error bounds
zbMATH Keywords: eigenfunctions
zbMATH Keywords: elliptic eigenvalue problems of fourth order
zbMATH Keywords: finite element
zbMATH Keywords: eigenvalue inclusions
zbMATH Keywords: interval arithmetic
zbMATH Keywords: plate
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01781561
OpenAlex ID: W2022877721
description: scientific article; zbMATH DE number 938555