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- Bogoya, Manuel, et al.
(författare)
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Fine spectral estimates with applications to the optimally fast solution of large FDE linear systems
- 2022
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Ingår i: BIT Numerical Mathematics. - : Springer Nature. - 0006-3835 .- 1572-9125. ; 62:4, s. 1417-1431
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Tidskriftsartikel (refereegranskat)abstract
- In the present article we consider a type of matrices stemming in the context of the numerical approximation of distributed order fractional differential equations (FDEs). From one side they could look standard, since they are real, symmetric and positive definite. On the other hand they cause specific difficulties which prevent the successful use of classical tools. In particular the associated matrix-sequence, with respect to the matrix-size, is ill-conditioned and it is such that a generating function does not exists, but we face the problem of dealing with a sequence of generating functions with an intricate expression. Nevertheless, we obtain a real interval where the smallest eigenvalue belongs to, showing also its asymptotic behavior. We observe that the new bounds improve those already present in the literature and give more accurate pieces of spectral information, which are in fact used in the design of fast numerical algorithms for the associated large linear systems, approximating the given distributed order FDEs. Very satisfactory numerical results are presented and critically discussed, while a section with conclusions and open problems ends the current work.
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| 4. |
- Nguyen, Quoc Khanh, 1993-, et al.
(författare)
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Spectral analysis of the finite element matrices approximating 2D linearly elastic structures and multigrid proposals
- 2022
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Ingår i: Numerical Linear Algebra with Applications. - : John Wiley & Sons. - 1070-5325 .- 1099-1506.
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Tidskriftsartikel (refereegranskat)abstract
- Topology optimization aims to find the best material layout subject to given constraints. The so-called material distribution methods cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. When solving the governing equation using the finite element method, a large number of elements are used to discretize the design domain, and an element-wise constant function approximates the coefficient field in the considered design domain. This article presents a spectral analysis of the (large) coefficient matrices associated with the linear systems stemming from the finite element discretization of a linearly elastic problem for an arbitrary coefficient field. Based on the spectral information, we design a multigrid method which turns out to be optimal, in the sense that the (arithmetic) cost for solving the related linear systems, up to a fixed desired accuracy, is proportional to the matrix-vector cost, which is linear in the corresponding matrix size. The method is tested, and the numerical results are very satisfactory in terms of linear cost and number of iterations, which is bounded by a constant independent of the matrix size.
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| 5. |
- Nguyen, Quoc Khanh, 1993-, et al.
(författare)
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Spectral Analysis of the Finite Element Matrices Approximating 3D Linearly Elastic Structures and Multigrid Proposals
- 2022
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Ingår i: Mathematical and Computational Applications. - : MDPI. - 1300-686X .- 2297-8747. ; 27:5, s. 1-22
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Tidskriftsartikel (refereegranskat)abstract
- The so-called material distribution methods for topology optimization cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. In practice, the finite element method is typically used to approximate that kind of governing equations by using a large number of elements to discretize the design domain, and an element-wise constant function approximates the coefficient field in that domain. This paper presents a spectral analysis of the coefficient matrices associated with the linear systems stemming from the finite element discretization of a linearly elastic problem for an arbitrary coefficient field in three spatial dimensions. The given theoretical analysis is used for designing and studying an optimal multigrid method in the sense that the (arithmetic) cost for solving the problem, up to a fixed desired accuracy, is linear in the corresponding matrix size. Few selected numerical examples are presented and discussed in connection with the theoretical findings.
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- Rahla, Ryma Imene, et al.
(författare)
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Spectral analysis of Pk Finite Element matrices in the case of Friedrichs–Keller triangulations via Generalized Locally Toeplitz technology
- 2020
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Ingår i: Numerical Linear Algebra with Applications. - : Wiley. - 1070-5325 .- 1099-1506. ; 27:4
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Tidskriftsartikel (refereegranskat)abstract
- In the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(-a(x) backward difference center dot), with a continuous and positive over omega?, omega being an open and bounded subset of Rd, d >= 1. For the numerical approximation, we consider the classical Pk Finite Elements, in the case of Friedrichs-Keller triangulations, leading, as usual, to sequences of matrices of increasing size. The new results concern the spectral analysis of the resulting matrix-sequences in the direction of the global distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on omega=(0,1)(2) and we give a brief account in the more involved case of variable coefficients and more general domains. Tools are drawn from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz sequences. Numerical results are shown for a practical evidence of the theoretical findings.
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