Space-Time FE-DG Discretization of the Anisotropic Diffusion Equation in Any Dimension: The Spectral Symbol

Benedusi, Pietro; Garoni, Carlo; Krause, Rolf; Li, Xiaozhou; Serra‐Capizzano, Stefano

doi:10.1137/17m113527x

Cited by 16 publications

(22 citation statements)

References 20 publications

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“…This result was proved in [11,Theorem A.6] by a direct (complicated and cumbersome) approach. By following step by step the proof of Theorem 6.5, we can give an alternative (much more lucid and simpler) proof of [11,Theorem A.6] based on the theory of block GLT sequences. From this reformulation, it appears more clearly that the (singular value and spectral) symbol a(x)κ [p,k] (θ) consists of the following two "ingredients".…”

Section: S and We Definementioning

confidence: 84%

“…Such an extension is of the utmost importance in practical applications. In particular, it provides the necessary tools for computing the spectral distribution of block structured matrices arising from the discretization of systems of DEs [76,Section 3.3] and from the higher-order FE or discontinuous Galerkin (DG) approximation of scalar/vectorial DEs; see Section 1.3 and [11,43,54,57]. A few applications of the theory of block GLT sequences developed in [55,56] have been presented in [49,52].…”

Section: Contributions and Structure Of The Present Workmentioning

confidence: 99%

Section: S and We Definementioning

confidence: 99%

“…If we discretize (6.29) by the space-time higher-order FE-DG approximation technique considered in [11], then the resulting (normalized) FE-DG discretization matrices enjoy an asymptotic spectral distribution described by a (q +1)(p−k)×(q +1)(p−k) matrix-valued function. This result was proved in [11,Theorem A.6] by a direct (complicated and cumbersome) approach. By following step by step the proof of Theorem 6.5, we can give an alternative (much more lucid and simpler) proof of [11,Theorem A.6] based on the theory of block GLT sequences.…”

Section: S and We Definementioning

confidence: 99%

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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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Section: S and We Definementioning

confidence: 84%

Section: Contributions and Structure Of The Present Workmentioning

confidence: 99%

Section: S and We Definementioning

confidence: 99%

Section: S and We Definementioning

confidence: 99%

See 2 more Smart Citations

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

Barbarino¹,

Garoni²,

Serra‐Capizzano³

2020

etna

Self Cite

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“…Just as the latter, the theory of block GLT sequences has been devised in order to solve a specific application problem, namely the problem of computing/analyzing the spectral distribution of matrices arising from the numerical discretization of differential problems. In particular, this theory applies to block-structured matrices arising from either the discretization of systems of differential equations (DEs) or the higher-order finite element (FE) or discontinuous Galerkin (DG) approximation of both scalar and vectorial DEs; see [9,19,20]. More details on the theory of block GLT sequences and its applications can be found in [20,23].…”

mentioning

confidence: 99%

Block GLT Sequences: Matrix Functions and Engineering Application

Garoni

Serra‐Capizzano

2019

ELA

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The theory of block generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the spectral distribution of block-structured matrices arising from the discretization of differential problems, with a special reference to systems of differential equations (DEs) and to the higher-order finite element or discontinuous Galerkin approximation of both scalar and vectorial DEs. In the present paper, the theory of block GLT sequences is extended by proving that $\{f(A_n)\}_n$ is a block GLT sequence as long as $f$ is continuous and $\{A_n\}_n$ is a block GLT sequence formed by Hermitian matrices. It is also provided a relevant application of this result to the computation of the distribution of the numerical eigenvalues obtained from the higher-order isogeometric Galerkin discretization of second-order variable-coefficient differential eigenvalue problems (a topic of interest not only in numerical analysis but also in engineering).

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

Rahla

Serra‐Capizzano

Possio

2020

Numerical Linear Algebra App

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SummaryIn the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(−a(x)∇·), with a continuous and positive over , Ω being an open and bounded subset of , d≥1. For the numerical approximation, we consider the classical 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 Ω=(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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Space-Time FE-DG Discretization of the Anisotropic Diffusion Equation in Any Dimension: The Spectral Symbol

Cited by 16 publications

References 20 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

Block GLT Sequences: Matrix Functions and Engineering Application

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

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