Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra

Matthies, Gunar

ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 5, pp. 855-874.

We present families of scalar nonconforming finite elements of arbitrary order $r\ge 1$ with optimal approximation properties on quadrilaterals and hexahedra. Their vector-valued versions together with a discontinuous pressure approximation of order $r-1$ form inf-sup stable finite element pairs of order $r$ for the Stokes problem. The well-known elements by Rannacher and Turek are recovered in the case $r=1$. A numerical comparison between conforming and nonconforming discretisations will be given. Since higher order nonconforming discretisation on quadrilaterals and hexahedra have less unknowns and much less non-zero matrix entries compared to corresponding conforming methods, these methods are attractive for numerical simulations.

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