Stable finite element methods with divergence augmentation for the Stokes problem?

Abstract

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.