Applications of Mathematics, first online, pp. 1-22

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

Received August 27, 2019.   Published online October 21, 2020.

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.
Keywords:  bound on eigenvalues; preconditioning; elliptic differential equation
Classification MSC:  65F08, 65N30
DOI:  10.21136/AM.2020.0217-19

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Affiliations:   Martin Ladecký, Ivana Pultarová (corresponding author), Jan Zeman, Czech Technical University in Prague, Jugoslávských partyzánů 1580/3, 160 00 Praha 6, Czech Republic, e-mail:,,

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