Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods

Abstract

A linear full elliptic second order Partial Differential Equation (PDE), defined on a d-dimensional domain Ω, is approximated by the isogeometric Galerkin method based on uniform tensor-product Bsplines of degrees (p 1 ,. .. , p d). The considered approximation process leads to a d-level stiffness matrix, banded in a multilevel sense. This matrix is close to a d-level Toeplitz structure when the PDE coefficients are constant and the physical domain Ω is just the hypercube (0, 1) d without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiffness matrices has been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coefficients and an arbitrary domain Ω, parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiffness matrices. This symbol describes the asymptotic eigenvalue distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). The mathematical technique used for computing the symbol is based on the theory of Generalized Locally Toeplitz (GLT) sequences.

Are the eigenvalues of the B‐spline isogeometric analysis approximation of −Δu = λu known in almost closed form?

“…[14][15][16] In the present paper, motivated by the aforesaid interest, we perform a detailed spectral analysis of the matrices arising from the B-spline IgA discretization of the Laplacian eigenproblem −Δu = u. Our main results, which will be detailed in Section 1.2, complement those of other works [7][8][9][10][11][12][13] and deliver a fast (parallel) interpolation-extrapolation algorithm for computing the eigenvalues of the considered IgA matrices.…”

“…In particular, the spectral investigation of matrices arising from the IgA discretization of PDEs has become a topic of interest in the scientific community, mainly due to the superiority of IgA over the classical finite element analysis (FEA) in approximating the spectrum of the underlying differential operator. [2][3][4][5][6] It is also worth recalling that recent spectral distribution results for IgA discretization matrices [7][8][9][10][11][12][13] turned out to be the keystone for designing fast IgA solvers. [14][15][16] In the present paper, motivated by the aforesaid interest, we perform a detailed spectral analysis of the matrices arising from the B-spline IgA discretization of the Laplacian eigenproblem −Δu = u.…”

Non‐Hermitian perturbations of Hermitian matrix‐sequences and applications to the spectral analysis of the numerical approximation of partial differential equations

“…We remind that the knowledge of the spectral symbol has a practical impact in obtaining fine estimates on the convergence speed of Krylov methods, when we face the problem of the efficient computation of the solution of large linear systems. Furthermore, especially in the context of generalized locally Toeplitz (GLT) matrix‐sequences arising in the approximation of PDEs, the computation and analysis of the spectral symbol have been used for designing efficient solvers combining preconditioning and multigrid/multiiterative methods (see References and references therein).…”

Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

“…The experience reveals that virtually any kind of numerical methods for the discretization of DEs gives rise to structured matrices A n whose asymptotic spectral distribution, as the mesh fineness parameter n tends to infinity, can be computed through the theory of GLT sequences. We refer the reader to ( [13] Section 10.5), ([14] Section 7.3), and [15,16,18] for applications of the theory of GLT sequences in the context of finite difference (FD) discretizations of DEs; to ( [13] Section 10.6), ( [14] Section 7.4), and [16,18,19] for the finite element (FE) case; to [20] for the finite volume (FV) case; to ( [13] Section 10.7), ( [14] Sections 7.5-7.7), and [21][22][23][24][25][26] for the case of isogeometric analysis (IgA) set", "measurable function", "a.e. ", etc.)…”