A nonconforming P₃ finite element is constructed by enriching the conforming P₃ finite element space with three P₃ nonconforming bubbles and six additional P₄ nonconforming bubbles, on each tetrahedron. Here the divergence of the P₄ bubble is not a P₃ polynomial, but a P₂ polynomial. This nonconforming P₃ finite element, combined with the discontinuous P₂ finite element, is inf-sup stable for solving the Stokes equations on general tetrahedral grids. Consequently such a mixed finite element method produces quasi-optimal solutions for solving the stationary Stokes equations. With these special P₄ bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory.
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Xu et al. (2024) studied this question.
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