The mixed finite element approximation scheme with divergence augmentation for the Stokes problem is analyzed. We show that the Pk+1 - Pk-1 triangular elements or the Q(k+1) - Q(k-1) quadrilateral elements in R-2, k greater than or equal to 1, are stable with h(k+1/2) convergence in H-1-norm for velocity and h(k) convergence in L-2-norm for pressure. Moreover, h(k+1) convergence in H(div)-norm for velocity can be shown if the domain is convex. In R-3, the cross-grid Pk+1 - Pk-1 tetrahedral elements, k greater than or equal to 2, can be analyzed analogously for the approximation scheme with divergence augmentation and pressure stabilization. A numerical test which confirms the convergence analysis is presented. (C) 2001 Elsevier Science Ltd. All rights reserved.