@TechReport{ it:2015-012,
author = {Carlo Garoni and Carla Manni and Stefano Serra-Capizzano
and Debora Sesana and Hendrik Speleers},
title = {Lusin Theorem, {GLT} Sequences and Matrix Computations: An
Application to the Spectral Analysis of {PDE}
Discretization Matrices},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2015},
number = {2015-012},
month = mar,
abstract = {We extend previous results on the spectral distribution of
discretization matrices arising from B-spline Isogeometric
Analysis (IgA) approximations of a general $d$-dimensional
second-order elliptic Partial Differential Equation (PDE)
with variable coefficients.
First, we provide the spectral symbol of the Galerkin
B-spline IgA stiffness matrices, assuming only that the PDE
coefficients belong to $L^{\infty}$. This symbol describes
the asymptotic spectral distribution when the fineness
parameters tend to zero (so that the matrix-size tends to
infinity).
Second, we prove the positive semi-definiteness of the
$d\times d$ symmetric matrix in the Fourier variables
$(\theta_1,\ldots,\theta_d)$, which appears in the
expression of the symbol. This matrix is related to the
discretization of the (negative) Hessian operator, and its
positive semi-definiteness implies the non-negativity of
the symbol.
The mathematical arguments used in our derivation are based
on the Lusin theorem, on the theory of Generalized Locally
Toeplitz (GLT) sequences, and on careful Linear Algebra
manipulations of matrix determinants.
These arguments are very general and can be also applied to
other PDE discretization techniques than B-spline IgA.}
}