Stable finite element methods with divergence augmentation for the stokes problem

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 Qk+1 − Qk−1 quadrilateral elements in R2, k ≥ 1, are stable with hk+View the MathML source convergence in H1-norm for velocity and hk convergence in L2-norm for pressure. Moreover, hk+1 convergence in H(div)-norm for velocity can be shown if the domain is convex. In R3, the cross-grid Pk+1 − Pk−1 tetrahedral elements, k ≥ 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.