A Numerical Existence Proof of Nodal Lines for the First Eigenfunction of the Plate Equation (1996)
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BibTeX
@MISC{Wieners96anumerical,
author = {Christian Wieners},
title = {A Numerical Existence Proof of Nodal Lines for the First Eigenfunction of the Plate Equation},
year = {1996}
}
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Abstract
. 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 \Gamma2 (\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. AMS Symbol classification: 65N25 Key words: biharmonic operator, eigenfunction enclosures, plate equation, nodal lines, embedding constants The numerical computations in Bauer-Reiss [2] and Hackbusch-Hoffmann [12] strongly indicate the existence of nodal lines for the first eigenfunction of the clamped plate \Delta 2 u \Gamma u = 0; u 2 H 2 0 (\Omega ); \Omega = (0; 1) 2 : Here, ...
