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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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| Published in: | Numerical linear algebra with applications 2021-10, Vol.28 (5), p.n/a |
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| container_title | Numerical linear algebra with applications |
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| creator | Pultarová, Ivana Ladecký, Martin |
| description | 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. |
| doi_str_mv | 10.1002/nla.2382 |
| format | article |
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| ispartof | Numerical linear algebra with applications, 2021-10, Vol.28 (5), p.n/a |
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| language | eng |
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| 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 |
| title | Two‐sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems |
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