Numerical Algorithms 32: 185–191, 2003.  2003 Kluwer Academic Publishers. Printed in the Netherlands. Upper bound of the constant in strengthened C.B.S. inequality for systems of linear partial differential equations B. Achchab a,b , S. Achchab b , O. Axelsson c and A. Souissi b,d a Université Hassan 1 er , Faculté des sciences juridiques, économiques et sociales, Settat, B.P. 784, Settat, Maroc b LERMA, Ecole Mohammadia d’Ingénieurs, Avenue Ibn Sina, B.P. 765, Rabat-Agdal, Maroc E-mail: achchab@yohoo.fr c Department of Mathematics, University of Nijmegen, Postbus 9010 NL-6500 GL Nijmegen, The Netherlands d Université Mohammed V-Agdal, Faculté des sciences, Département de Mathématiques et d’Informatique, Avenue Ibn Battouta, B.P. 1014, Rabat, Maroc Received 20 February 2002; accepted 6 November 2002 Communicated by C. Brezinski The constant γ in the strengthened Cauchy–Bunyakowski–Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is the framework of finite element approxima- tions of systems of partial differential equations. We consider an approximation of general systems of linear partial differential equations in R 3 . Concerning a multilevel convergence rate corresponding to nested general tetrahedral meshes of size h and 2h, we give an estimate of this constant for general three-dimensional cases. Keywords: systems of PDE, finite element methods, preconditioning, multilevel methods, C.B.S. inequality 1. Introduction In recent years, multigrid and multilevel methods have been successfully applied to the treatment of the linear equations that arise from the discretization of partial differ- ential equations, in particular the elliptic ones. These methods have been proven to have an optimal or nearly optimal complexity for solution of discrete systems, because they are based on discretization hierarchies obtained from successively refined meshes, and they can be embedded especially well in an adaptive framework.