Finite element methods for the Stokes system based on a Zienkiewicz type N-simplex

• New FEM for 2D and 3D Stokes system with Hermite velocity of the Zienkiewicz type.
• Optimal error estimates for Galerkin formulation in 2D with criss-cross meshes.
• Error estimates for GLS formulation bypassing Clément or Scott–Zhang interpolations.
• Inf–sup free GLS error analysis in terms of inhomogeneous boundary velocity.
• Comparative numerics with Lagrange pair of same order.

Hermite interpolation is increasingly showing to be a powerful numerical solution tool, as applied to different kinds of second order boundary value problems. In this work we present two Hermite finite element methods to solve viscous incompressible flows problems, in both two- and three-dimension space. In the two-dimensional case we use the Zienkiewicz triangle to represent the velocity field, and in the three-dimensional case an extension of this element to tetrahedra, still called a Zienkiewicz element. Taking as a model the Stokes system, the pressure is approximated with continuous functions, either piecewise linear or piecewise quadratic, according to the version of the Zienkiewicz element in use, that is, with either incomplete or complete cubics. The methods employ both the standard Galerkin or the Petrov–Galerkin formulation first proposed in Hughes et al. (1986) [18], based on the addition of a balance of force term. A priori error analyses point to optimal convergence rates for the PG approach, and for the Galerkin formulation too, at least in some particular cases. From the point of view of both accuracy and the global number of degrees of freedom, the new methods are shown to have a favorable cost-benefit ratio, as compared to velocity Lagrange finite elements of the same order, especially if the Galerkin approach is employed.