@TechReport{	  it:2002-039,
  author	= {R. Blaheta and S. Margenov and M. Neytcheva},
  title		= {Uniform estimate of the constant in the strengthened {CBS}
		  inequality for anisotropic non-conforming {FEM} systems},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2002},
  number	= {2002-039},
  month		= nov,
  abstract	= {Preconditioners based on various multilevel extensions of
		  two-level finite element methods (FEM) lead to iterative
		  methods which have an optimal order computational
		  complexity with respect to the size of the system. Such
		  methods were first presented in \cite{AV1,AV2}, and are
		  based on (recursive) two-level splittings of the finite
		  element space. The key role in the derivation of optimal
		  convergence rate estimates plays the constant $\gamma$ in
		  the so-called Cauchy-Bunyakowski-Schwarz (CBS) inequality,
		  associated with the angle between the two subspaces of the
		  splitting. It turns out that only existence of uniform
		  estimates for this constant is not enough and accurate
		  quantitative bounds for $\gamma$ have to be found as well.
		  More precisely, the value of the upper bound for $\gamma\in
		  (0,1)$ is a part of the construction of various multilevel
		  extensions of the related two-level methods.
		  
		  In this paper an algebraic two-level preconditioning
		  algorithm for second order elliptic boundary value problems
		  is constructed, where the discretization is done using
		  Crouzeix-Raviart non-conforming linear finite elements on
		  triangles. An important point to make is that in this case
		  the finite element spaces corresponding to two successive
		  levels of mesh refinements are not nested. To handle this,
		  a proper two-level basis is considered, which enables us to
		  fit the general framework for the construction of two-level
		  preconditioners for conforming finite elements and to
		  generalize the method to the multilevel case.
		  
		  The major contribution of this paper is the derived
		  estimates of the related constant $\gamma$ in the
		  strengthened CBS inequality. These estimates are uniform
		  with respect to both coefficient and mesh anisotropy. Up to
		  our knowledge, the results presented in the paper are the
		  first for non-conforming FEM systems. }
}