Cauchy-Bunyakowski-Schwarz inequality; multilevel preconditioning; elliptic partial differential equation
We estimate the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for hierarchical bilinear finite element spaces and elliptic partial differential equations with coefficients corresponding to anisotropy (orthotropy). It is shown that there is a nontrivial universal estimate, which does not depend on anisotropy. Moreover, this estimate is sharp and the same as for hierarchical linear finite element spaces.
 O. Axelsson, R. Blaheta: Two simple derivations of universal bounds for the C.B.S. inequality constant
. Applications of Mathematics (to appear). MR 2032148
 R. Blaheta: GPCG-generalized preconditioned CD method and its use with non-linear and non-symmetric displacement decomposition preconditioners
. Numer. Linear Algebra Appl. 9 (2002), 527–550. DOI 10.1002/nla.295
| MR 1934875
 O. Axelsson, V. A. Barker: Finite element solution of boundary value problems: Theory and computations
. Classics in Appl. Math, SIAM, Philadelphia, 2001. MR 1856818
 J. F. Maitre, F. Mussy: The contraction number of a class of twolevel methods, an exact evaluation for some finite element subspaces and model problem. In: Multigrid Methods, Lecture Notes in Math. 960, W. Hackbusch, U. Trottenberg (eds.), Springer-Verlag, Berlin, 1982, pp. 535–544.