@TechReport{ it:2014-021,
author = {Carlo Garoni and Stefano Serra-Capizzano and Debora
Sesana},
title = {Spectral Analysis and Spectral Symbol of $d$-variate
$\mathbb{Q}_p$ {L}agrangian {FEM} Stiffness Matrices},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2014},
number = {2014-021},
month = nov,
abstract = {We study the spectral properties of the stiffness matrices
coming from the $\mathbb{Q}_p$ Lagrangian FEM approximation
of $d$-dimensional second order elliptic differential
problems; here, $p=(p_1,\ldots,p_d)\in\mathbb{N}^d$ and
$p_j$ represents the polynomial approximation degree in the
$j$-th direction. After presenting a construction of these
matrices, we investigate the conditioning (behavior of the
extremal eigenvalues and singular values) and the
asymptotic spectral distribution in the Weyl sense, and we
find out the so-called (spectral) symbol describing the
asymptotic spectrum.
We also study the properties of the symbol, which turns out
to be a $d$-variate function taking values in the space of
$D(p)\times D(p)$ Hermitian matrices, where
$D(p)=\prod_{j=1}^d p_j$. Unlike the stiffness matrices
coming from the $p$\,-degree B-spline IgA approximation of
the same differential problems, where a unique $d$-variate
real-valued function describes all the spectrum, here the
spectrum is described by $D(p)$ different functions, that
is the $D(p)$ eigenvalues of the symbol, which are
well-separated, far away, and exponentially diverging with
respect to $p$ and $d$. This very involved picture provides
a clean explanation of: a) the difficulties encountered in
designing robust solvers, with convergence speed
independent of the matrix size, of the approximation
parameters $p$, and of the dimensionality $d$; b) the
possible convergence deterioration of known iterative
methods, already for moderate $p$ and $d$.}
}