Error Estimates in the Fixed Membrane Problem

2010 ◽  
pp. 231-244
Author(s):  
B. E. Hubbard
Keyword(s):  
Error Estimates ◽  
Fixed Membrane ◽  
Membrane Problem

THE RANGE OF VALUES OF λ2/λ1 AND λ3/λ1 FOR THE FIXED MEMBRANE PROBLEM

1994 ◽  
Vol 06 (05a) ◽  
pp. 999-1009 ◽  
Author(s):  
MARK S. ASHBAUGH ◽  
RAFAEL D. BENGURIA
Keyword(s):  
Bounded Domain ◽  
The Other ◽  
Dirichlet Laplacian ◽  
Fixed Membrane ◽  
De Vries ◽  
Membrane Problem ◽  
Range Of Values

We investigate the region of the plane in which the point (λ2/λ1, λ3/λ1) can lie, where λ1, λ2, and λ3 are the first three eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain Ω ⊂ ℝ2. In particular, by making use of a technique introduced by de Vries we obtain the best bounds to date for the quantities λ3/λ1 and (λ2 + λ3)/λ1. These bounds are λ3/λ1 ≤ 3.90514+ and (λ2 + λ3)/λ1 ≤ 5.52485+ and give small improvements over previous bounds of Marcellini. Where Marcellini used a bound due to Brands in his argument we use a better version of this bound which we obtain by incorporating deVries' idea. The other bounds that yield the greatest information about the region where points (λ2/λ1, λ3/λ1) can (possibly) lie are those due to Marcellini, Hile and Protter, and us (of which there are several, with two of them being new with this paper).

Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry

2019 ◽  
Vol 19 (3) ◽  
pp. 415-430 ◽  
Author(s):  
Fleurianne Bertrand ◽  
Zhiqiang Cai ◽  
Eun Young Park
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
Error Estimates ◽  
Stokes Equations ◽  
Displacement Velocity ◽  
Three Dimensions ◽  
Optimal Error Estimates ◽  
Finite Element Approximations ◽  
Solution Methods ◽  
Membrane Problem

AbstractThis paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the{L^{2}}norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the norm of{H(\mathrm{div};\Omega)}for the stress, of{H^{1}(\Omega)}for the displacement/velocity, and of{L^{2}(\Omega)}for the rotation/vorticity and the pressure. This immediately implies optimal error estimates in the energy norm for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart–Thomas finite element spaces are used to approximate the stress tensor. Through a refined duality argument, an optimal{L^{2}}norm error estimates for the displacement/velocity are also established. Finally, numerical results for a Cook’s membrane problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit.

Isoperimetric bounds for higher eigenvalue ratios for the n-dimensional fixed membrane problem

1993 ◽  
Vol 123 (6) ◽  
pp. 977-985 ◽  
Author(s):  
Mark S. Ashbaugh ◽  
Rafael D. Benguria
Keyword(s):  
Bessel Function ◽  
Bounded Domain ◽  
Eigenvalue Problems ◽  
Positive Zero ◽  
Dirichlet Laplacian ◽  
Nodal Domains ◽  
Good Bound ◽  
Fixed Membrane ◽  
Membrane Problem ◽  
Elliptic Eigenvalue Problems

SynopsisWe give several results which extend our recent proof of the Payne-Pólya–Weinberger conjecture to ratios of higher eigenvalues. In particular, we show that for a bounded domain Ω⊂ℝn the eigenvalues of its Dirichlet Laplacian obey where λm denotes the mth eigenvalue and jp,k denotes the kth positive zero of the Bessel function Jp(x). Certain extensions of this result are given, the most general being the bound where k≧2 and l(m) denotes the number of nodal domains of an mth eigenfunction. Our results imply certain further conjectures of Payne, Pólya, and Weinberger concerning λ3/λ2 and λ4/λ3. In addition, we find a resonably good bound on λ4/λ1. We also briefly discuss extensions to Schrödinger operators and other elliptic eigenvalue problems.

Systematic effects in observed parallaxes

1978 ◽  
Vol 48 ◽  
pp. 31-35
Author(s):  
R. B. Hanson
Keyword(s):  
Error Estimates ◽  
Zero Point ◽  
Absolute Zero ◽  
Apparent Magnitude ◽  
Coherent System ◽  
Preliminary Results ◽  
General Catalogue ◽  
Systematic Effects ◽  
New Evidence ◽  
Right Ascension

Several outstanding problems affecting the existing parallaxes should be resolved to form a coherent system for the new General Catalogue proposed by van Altena, as well as to improve luminosity calibrations and other parallax applications. Lutz has reviewed several of these problems, such as: (A) systematic differences between observatories, (B) external error estimates, (C) the absolute zero point, and (D) systematic observational effects (in right ascension, declination, apparent magnitude, etc.). Here we explore the use of cluster and spectroscopic parallaxes, and the distributions of observed parallaxes, to bring new evidence to bear on these classic problems. Several preliminary results have been obtained.

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