@TechReport{ it:2015-008,
author = {Ali Dorostkar and Maya Neytcheva and Stefano
Serra-Capizzano},
title = {Spectral Analysis of Coupled {PDE}s and of their {S}chur
Complements via the Notion of {G}eneralized {L}ocally
{T}oeplitz Sequences},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2015},
number = {2015-008},
month = feb,
abstract = {We consider large linear systems of algebraic equations
arising from the Finite Element approximation of coupled
partial differential equations. As case study we focus on
the linear elasticity equations, formulated as a saddle
point problem to allow for modeling of purely
incompressible materials. Using the notion of the so-called
\textit{spectral symbol} in the Generalized Locally
Toeplitz (GLT) setting, we derive the GLT symbol (in the
Weyl sense) of the sequence of matrices $\{A_n\}$
approximating the elasticity equations. Further, exploiting
the property that the GLT class { defines an algebra of
matrix sequences} and the fact that the Schur complements
are obtained via elementary algebraic operation on the
blocks of $A_n$, we derive the symbols $f^{\mathcal{S}}$ of
the associated sequences of Schur complements $\{S_n\}$. As
a consequence of the GLT theory, the eigenvalues of $S_n$
for large $n$ are described by a sampling of
$f^{\mathcal{S}}$ on a uniform grid of its domain of
definition. We extend the existing GLT technique with novel
elements, related to block-matrices and Schur complement
matrices, and illustrate the theoretical findings with
numerical tests. }
}