Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method

Martin Ladecký; Ivana Pultarová; Jan Zeman

Applications of Mathematics (2021)

  • Volume: 66, Issue: 1, page 21-42
  • ISSN: 0862-7940

Abstract

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A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, and preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data.

How to cite

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Ladecký, Martin, Pultarová, Ivana, and Zeman, Jan. "Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method." Applications of Mathematics 66.1 (2021): 21-42. <http://eudml.org/doc/296961>.

@article{Ladecký2021,
abstract = {A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, and preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data.},
author = {Ladecký, Martin, Pultarová, Ivana, Zeman, Jan},
journal = {Applications of Mathematics},
keywords = {bound on eigenvalues; preconditioning; elliptic differential equation},
language = {eng},
number = {1},
pages = {21-42},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method},
url = {http://eudml.org/doc/296961},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Ladecký, Martin
AU - Pultarová, Ivana
AU - Zeman, Jan
TI - Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 21
EP - 42
AB - A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, and preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data.
LA - eng
KW - bound on eigenvalues; preconditioning; elliptic differential equation
UR - http://eudml.org/doc/296961
ER -

References

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