Two‐sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems

Numerical methods for elliptic partial differential equations usually lead to systems of linear equations with sparse, symmetric, and positive definite matrices. In many methods, these matrices can be obtained as sums of local symmetric positive semidefinite matrices. In this article, we use this as...

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Bibliographic Details
Published in:Numerical linear algebra with applications 2021-10, Vol.28 (5), p.n/a
Main Authors: Pultarová, Ivana, Ladecký, Martin
Format: Article
Language:English
Subjects:
algebraic multilevel method
Discretization
eigenvalue bounds
Eigenvalues
Elliptic functions
Finite difference method
finite element method
Linear equations
Matrices (mathematics)
Numerical methods
Partial differential equations
preconditioning
stochastic Galerkin finite element method
Upper bounds
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Summary:Numerical methods for elliptic partial differential equations usually lead to systems of linear equations with sparse, symmetric, and positive definite matrices. In many methods, these matrices can be obtained as sums of local symmetric positive semidefinite matrices. In this article, we use this assumption and introduce a method that provides guaranteed lower and upper bounds to all individual eigenvalues of the preconditioned matrices. We apply the method for preconditioners arising from the same discretization problem but with simplified coefficients. The method uses solely the data over the solution domain and local connections between the degrees of freedom defined by the discretization.
ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2382