In this paper, we present energy estimates for the stresses and velocity components in a general setting, for both inertial and inertialess flows of an Oldroyd B fluid. Our results apply to flows in bounded domains in any number of dimensions, subject to Dirichlet and possibly inflow boundary conditions. A novel numerical scheme is introduced and shown to be superior to a conventional Galerkin discretization of the Oldroyd B equations. In particular, the new scheme respects the derived energy estimates and guarantees positive definiteness of the stress tensor tau + ((1 - beta)/We)I at all times, beta being a solvent-to-total viscosity ratio and We a Weissenberg number. Numerical results for the planar viscoelastic Poiseuille problem illustrate some differences between the new and conventional schemes and reveal that the conventional scheme may lead to violation of the theoretical energy bounds in certain circumstances. (C) 2003 Elsevier B.V. All rights reserved.