An optimal adaptive FEM for eigenvalue clusters

D. Gallistl

Research output: Contribution to journalArticleAcademicpeer-review

12 Citations (Scopus)

Abstract

The analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. A first analysis for multiple eigenvalues of the recent work Dai et al. (arXiv Preprint 1210.1846v2, 2013) introduces an adaptive method whose marking strategy is based on the element-wise sum of local error estimator contributions for multiple eigenvalues. This paper proves the optimality of a practical adaptive algorithm based on a lowest-order conforming finite element method for eigenvalue clusters for the eigenvalues of the Laplace operator in terms of nonlinear approximation classes. All estimates are explicit in the initial mesh-size, the eigenvalues and the cluster width to clarify the dependence of the involved constants.
Original languageEnglish
Pages (from-to)467-496
Number of pages30
JournalNumerische Mathematik
Volume130
Issue number3
DOIs
Publication statusPublished - 2015
Externally publishedYes

Fingerprint

Eigenvalue
Multiple Eigenvalues
Finite element method
Adaptive Finite Element Method
Adaptive algorithms
Nonlinear Approximation
Error Estimator
Laplace Operator
Adaptive Method
Adaptive Algorithm
Immediately
Lowest
Optimality
Finite Element Method
Mesh
Estimate
Class
Strategy

Cite this

Gallistl, D. / An optimal adaptive FEM for eigenvalue clusters. In: Numerische Mathematik. 2015 ; Vol. 130, No. 3. pp. 467-496.
@article{160e96ef3c424ef4bfb35214dcbd9062,
title = "An optimal adaptive FEM for eigenvalue clusters",
abstract = "The analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. A first analysis for multiple eigenvalues of the recent work Dai et al. (arXiv Preprint 1210.1846v2, 2013) introduces an adaptive method whose marking strategy is based on the element-wise sum of local error estimator contributions for multiple eigenvalues. This paper proves the optimality of a practical adaptive algorithm based on a lowest-order conforming finite element method for eigenvalue clusters for the eigenvalues of the Laplace operator in terms of nonlinear approximation classes. All estimates are explicit in the initial mesh-size, the eigenvalues and the cluster width to clarify the dependence of the involved constants.",
author = "D. Gallistl",
year = "2015",
doi = "10.1007/s00211-014-0671-8",
language = "English",
volume = "130",
pages = "467--496",
journal = "Numerische Mathematik",
issn = "0029-599X",
publisher = "Springer",
number = "3",

}

An optimal adaptive FEM for eigenvalue clusters. / Gallistl, D.

In: Numerische Mathematik, Vol. 130, No. 3, 2015, p. 467-496.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - An optimal adaptive FEM for eigenvalue clusters

AU - Gallistl, D.

PY - 2015

Y1 - 2015

N2 - The analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. A first analysis for multiple eigenvalues of the recent work Dai et al. (arXiv Preprint 1210.1846v2, 2013) introduces an adaptive method whose marking strategy is based on the element-wise sum of local error estimator contributions for multiple eigenvalues. This paper proves the optimality of a practical adaptive algorithm based on a lowest-order conforming finite element method for eigenvalue clusters for the eigenvalues of the Laplace operator in terms of nonlinear approximation classes. All estimates are explicit in the initial mesh-size, the eigenvalues and the cluster width to clarify the dependence of the involved constants.

AB - The analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. A first analysis for multiple eigenvalues of the recent work Dai et al. (arXiv Preprint 1210.1846v2, 2013) introduces an adaptive method whose marking strategy is based on the element-wise sum of local error estimator contributions for multiple eigenvalues. This paper proves the optimality of a practical adaptive algorithm based on a lowest-order conforming finite element method for eigenvalue clusters for the eigenvalues of the Laplace operator in terms of nonlinear approximation classes. All estimates are explicit in the initial mesh-size, the eigenvalues and the cluster width to clarify the dependence of the involved constants.

U2 - 10.1007/s00211-014-0671-8

DO - 10.1007/s00211-014-0671-8

M3 - Article

VL - 130

SP - 467

EP - 496

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 3

ER -