@TechReport{ it:2015-005,
author = {Carlo Garoni and Carla Manni and Stefano Serra-Capizzano
and Debora Sesana and Hendrik Speleers},
title = {Spectral Analysis and Spectral Symbol of Matrices in
Isogeometric {G}alerkin Methods},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2015},
number = {2015-005},
month = jan,
abstract = {A linear full elliptic second order Partial Differential
Equation (PDE), defined on a $d$-dimensional domain
$\Omega$, is approximated by the isogeometric Galerkin
method based on uniform tensor-product B-splines of degrees
$(p_1,\ldots,p_d)$. 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 $\Omega$ 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
$\Omega$, 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.}
}