@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. }
}