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Crouzeix-Velte decompositions for higher-order finite elements. (English) Zbl 1136.65110

Summary: We consider polynomial finite elements of order \(k \geq 1\) 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 \((k = 1)\) [cf. M. Crouzeix and P.-A. Raviart, Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973; Zbl 0302.65087)], Fortin-Soulie \((k = 2)\) [cf. M. Fortin and M. Soulie, Int. J. Numer. Methods Eng. 19, 505–520 (1983; Zbl 0514.73068)] and Crouzeix-Falk \((k = 3)\) elements [cf. M. Crouzeix and R. S. Falk, Math. Comput. 52, No. 186, 437–456 (1989; Zbl 0685.76018)].
We show that, for odd orders, these Gauss-Legendre elements do not possess a Crouzeix-Velte decomposition [cf. M. Crouzeix, IRIA Cahier, No. 12 (1974); W. Velte, Lect. Notes Math. 1431, 158-168 (1990; Zbl 0707.35107)]. 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 [cf. L. R. Scott and M. Vogelius, Lect. Appl. Math. 22, Pt. 2, 221–244 (1985; Zbl 0582.76028] is avoided by these elements which are shown to differ from the former ones by nonconforming bubbles. We also consider quadrilateral elements of order \(k \geq 1\) where the requirement of a Crouzeix-Velte decomposition is shown to exclude most commonly used elements.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
35J25 Boundary value problems for second-order elliptic equations
74B05 Classical linear elasticity
74S05 Finite element methods applied to problems in solid mechanics
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