Carlo Garoni

Spectral analysis and spectral symbol of matrices in isogeometric collocation methods

A linear full elliptic second order Partial Dierential Equation (PDE), defined on a d-dimensional domain , is approximated by the isogeometric Galerkin method based on uniform tensor-product Bsplines of degrees (p1;:::; pd). The considered approximation process leads to a d-level stiness matrix, banded in a multilevel sense. This matrix is close to a d-level Toeplitz structure when the PDE coecients are constant and the physical domain is just the hypercube (0; 1) d without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiness matrices has been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coecients and an arbitrary domain , parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiness 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.

The theory of generalized locally Toeplitz sequences : a review, an extension, and a few representative applications

We review and extend the theory of Generalized Locally Toeplitz (GLT) sequences, which goes back to Tilli’s work on Locally Toeplitz sequences and was developed by the second author during the last decade. Informally speaking, a GLT sequence {A n } n is a sequence of matrices with increasing size equipped with a function κ (the so-called symbol). We write {A n } n ~glt κ to indicate that {A n } n is a GLT sequence with symbol κ. This symbol characterizes the asymptotic singular value distribution of {A n } n ; if the matrices A n are Hermitian, it also characterizes the asymptotic eigenvalue distribution of {A n } n . Three fundamental examples of GLT sequences are: (i) the sequence of Toeplitz matrices generated by a function f in L 1; (ii) the sequence of diagonal sampling matrices containing the samples of a Riemann-integrable function a over equispaced grids; (iii) any zero-distributed sequence, i.e., any sequence of matrices with an asymptotic singular value distribution characterized by 0. The symbol of the GLT sequence (i) is f, the symbol of the GLT sequence (ii) is a, and the symbol of the GLT sequences (iii) is 0. The set of GLT sequences is a *-algebra. More precisely, suppose that {A n (i) } n ~glt κ i for i = 1, … ,r, and let A n = ops(A n (1) , … , A n (r) ) be a matrix obtained from A n (1) , … , A n (r) by means of certain algebraic operations “ops”, such as linear combinations, products, inversions and conjugate transpositions; then {A n } n ~glt k = ops(k 1, … , k r ).

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

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 differential equations (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 DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations.

Two-grid optimality for Galerkin linear systems based on B-splines

A multigrid method for linear systems stemming from the Galerkin B-spline discretization of classical second-order elliptic problems is considered. The spectral features of the involved stiffness matrices, as the fineness parameter h tends to zero, have been deeply studied in previous works, with particular attention to the dependencies of the spectrum on the degree p of the B-splines used in the discretization process. Here, by exploiting this information in connection with

The theory of locally Toeplitz sequences: a review, an extension, and a few representative applications

\tau

Discontinuous Galerkin discretization of the heat equation in any dimension : The spectral symbol

τ-matrices, we describe a multigrid strategy and we prove that the corresponding two-grid iterations have a convergence rate independent of h for

On the asymptotic spectrum of stiffness matrices arising from IgA

p=1,2,3