@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.}
}