Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators

Andrey Andreev; Milena Racheva

Applications of Mathematics (2014)

  • Volume: 59, Issue: 4, page 371-390
  • ISSN: 0862-7940

Abstract

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This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues λ l , l = 2 , 3 , ... , we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method.

How to cite

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Andreev, Andrey, and Racheva, Milena. "Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators." Applications of Mathematics 59.4 (2014): 371-390. <http://eudml.org/doc/261879>.

@article{Andreev2014,
abstract = {This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues $\lambda _l$, $l = 2,3,\ldots $, we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method.},
author = {Andreev, Andrey, Racheva, Milena},
journal = {Applications of Mathematics},
keywords = {eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound; eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound; two-sided bounds; fourth-order problems; numerical experiments; plate bending},
language = {eng},
number = {4},
pages = {371-390},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators},
url = {http://eudml.org/doc/261879},
volume = {59},
year = {2014},
}

TY - JOUR
AU - Andreev, Andrey
AU - Racheva, Milena
TI - Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators
JO - Applications of Mathematics
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 4
SP - 371
EP - 390
AB - This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues $\lambda _l$, $l = 2,3,\ldots $, we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method.
LA - eng
KW - eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound; eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound; two-sided bounds; fourth-order problems; numerical experiments; plate bending
UR - http://eudml.org/doc/261879
ER -

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