Spectral analysis of Pk Finite Element matrices in the case of Friedrichs–Keller triangulations via Generalized Locally Toeplitz technology

Rahla, Ryma Imene; Serra‐Capizzano, Stefano; Possio, Cristina Tablino

doi:10.1002/nla.2302

Cited by 6 publications

(7 citation statements)

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“…In particular, block GLT sequences are encountered in the discretization of vectorial DEs (systems of scalar DEs) as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. We refer the reader to [50,Section 10.5], [51,Section 7.3], and [20,49,75,76] for applications of the theory of GLT sequences in the context of finite difference (FD) discretizations of DEs; to [50,Section 10.6], [51,Section 7.4], and [10,20,42,49,57,67,76] for the finite element (FE) case; to [12] for the finite volume (FV) case; to [50,Section 10.7], [51,, and [36,45,46,47,48,52,57,68] for the case of isogeometric analysis (IgA) discretizations, both in the collocation and Galerkin frameworks; and to [40] for a further application to fractional DEs. We also refer the reader to [50,Section 10.4] and [1,72] for a look at the GLT approach for sequences of matrices arising from IE discretizations.…”

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confidence: 99%

Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case

Barbarino¹,

Garoni²,

Serra‐Capizzano³

2020

etna

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In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numerical discretization of a differential equation (DE), knowledge of the spectral distribution of the associated matrix has proved to be useful information for designing/analyzing appropriate solvers-especially, preconditioned Krylov and multigrid solvers-for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material). The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of DEs. Indeed, when the mesh-fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}n, which often turns out to be a GLT sequence or one of its "relatives", i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar/vectorial DEs. This work is a review, refinement, extension, and systematic exposition of the theory of block GLT sequences. It also includes several emblematic applications of this theory in the context of DE discretizations. Key words. asymptotic distribution of singular values and eigenvalues, block Toeplitz matrices, block generalized locally Toeplitz matrices, numerical discretization of differential equations, finite differences, finite elements, isogeometric analysis, discontinuous Galerkin methods, tensor products, B-splines.

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confidence: 99%

Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case

Barbarino¹,

Garoni²,

Serra‐Capizzano³

2020

etna

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“…Furthermore, a generalization of the previous result in higher dimension is given in [8] and is reported in the subsequent theorem.…”

Section: Structure Of the Matrices And Spectral Analysismentioning

confidence: 60%

“…We remind that a similar analysis is carried out in [8] for the finite approximations P k for k ≥ 2 and for d = 2: the analysis for d = 1 is contained in [3] trivially because Q k ≡ P k for every k ≥ 1, while, for d = 2, and even more for d ≥ 3, the situation is greatly complicated by the fact that we do not encounter a tensor structure. Nevertheless, the picture is quite similar and the obtained information in terms of spectral symbol is sufficient for deducing a quite accurate analysis concerning the distribution and the extremal behavior of the eigenvalues of the resulting matrix-sequences.…”

Section: Introductionmentioning

confidence: 98%

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Multigrid for Q k Finite Element Matrices Using a (Block) Toeplitz Symbol Approach

et al. 2019

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In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the Q k Finite Elements approximation of one- and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div − a ( x ) ∇ · , with a continuous and positive over Ω ¯ , Ω being an open and bounded subset of R 2 . While the analysis is performed in one dimension, the numerics are carried out also in higher dimension d ≥ 2 , showing an optimal behavior in terms of the dependency on the matrix size and a substantial robustness with respect to the dimensionality d and to the polynomial degree k.

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“…Experience reveals that virtually any kind of numerical method for the discretization of PDEs gives rise to structured matrices A n whose asymptotic spectral distribution, as the meshfineness parameter n tends to infinity, can be computed through the theory of GLT sequences. We refer the reader to [40,Section 10.5], [41,Section 7.3], and [16,62,63] for applications of the theory of GLT sequences in the context of finite difference (FD) discretizations of PDEs; to [40,Section 10.6], [41,Section 7.4], and [11,16,33,55,63] for the finite element (FE) case; to [13] for the finite volume (FV) case; to [40,Section 10.7], [41,, and [27,35,36,37,38,56] for the case of isogeometric analysis (IgA) discretizations, both in the collocation and Galerkin frameworks; and to [31] for a further application to fractional differential equations. We also refer the reader to [40,Section 10.4] and [1,59] for a look at the GLT approach for sequences of matrices arising from IE discretizations.…”

mentioning

confidence: 99%

Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case

Barbarino¹,

Garoni²,

Serra‐Capizzano³

2020

etna

View full text Add to dashboard Cite

In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numerical discretization of a partial differential equation (PDE), knowledge of the spectral distribution of the associated matrix has proved to be useful information for designing/analyzing appropriate solvers-especially, preconditioned Krylov and multigrid solvers-for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material). The theory of multilevel generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of PDEs. Indeed, when the mesh-fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}n, which often turns out to be a multilevel GLT sequence or one of its "relatives", i.e., a multilevel block GLT sequence or a (multilevel) reduced GLT sequence. In particular, multilevel block GLT sequences are encountered in the discretization of systems of PDEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar/vectorial PDEs. In this work, we systematically develop the theory of multilevel block GLT sequences as an extension of the theories of (unilevel) GLT sequences [Garoni

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Spectral analysis of Pk Finite Element matrices in the case of Friedrichs–Keller triangulations via Generalized Locally Toeplitz technology

Cited by 6 publications

References 43 publications

Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case

Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case

Multigrid for Q k Finite Element Matrices Using a (Block) Toeplitz Symbol Approach

Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case

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