Crouzeix-Velte decompositions for higher-order finite elements

We consider polynomial finite elements of order k >= 1 for the approximation of Stokes and linear elasticity problems which are continuous in the Gauss-Legendre points of the elements sides, i.e., generalize the Crouzeix-Raviart (k = 1), Fortin-Soulie (k = 2) and Crouzeix-Falk (k = 3) elements. We show that, for odd orders, these Gauss-Legendre elements do not possess a Crouzeix-Velte decomposition. For even orders, not only a Crouzeix-Velte decomposition can be shown to exist (which is advantageous when solving the corresponding linear equations and eigenvalue problems) but also the grid singularity of the well-known Scott-Vogelius elements is avoided by these elements which are shown to differ from the former ones by nonconforming bubbles. We also consider quadrilateral elements of order k >= 1 where the requirement of a Crouzeix-Velte decomposition is shown to exclude most commonly used elements.