@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. }
}