Geodesic
Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method
Applications of Mathematics, Tome 66 (2021) no. 1, pp. 21-42 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/AM.2020.0217-19
Classification : 65F08, 65N30
Keywords: bound on eigenvalues; preconditioning; elliptic differential equation 
@article{10_21136_AM_2020_0217_19,
     author = {Ladeck\'y, Martin and Pultarov\'a, Ivana and Zeman, Jan},
     title = {Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method},
     journal = {Applications of Mathematics},
     pages = {21--42},
     year = {2021},
     volume = {66},
     number = {1},
     doi = {10.21136/AM.2020.0217-19},
     mrnumber = {4218600},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.2020.0217-19/}
}
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
SP  - 21
EP  - 42
VL  - 66
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/AM.2020.0217-19/
DO  - 10.21136/AM.2020.0217-19
LA  - en
ID  - 10_21136_AM_2020_0217_19
ER  - 
%0 Journal Article
%A Ladecký, Martin
%A Pultarová, Ivana
%A Zeman, Jan
%T Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method
%J Applications of Mathematics
%D 2021
%P 21-42
%V 66
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.2020.0217-19/
%R 10.21136/AM.2020.0217-19
%G en
%F 10_21136_AM_2020_0217_19
Ladecký, Martin; Pultarová, Ivana; Zeman, Jan. Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method. Applications of Mathematics, Tome 66 (2021) no. 1, pp. 21-42. doi: 10.21136/AM.2020.0217-19

Cité par Sources :