A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids
<|reference_start|>A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids: A nonconforming $P_3$ finite element is constructed by enriching the conforming $P_3$ finite element space with three $P_3$ nonconforming bubbles and six additional $P_4$ nonconforming bubbles, on each tetrahedron. Here the divergence of the $P_4$ bubble is not a $P_3$ polynomial, but a $P_2$ polynomial. This nonconforming $P_3$ finite element, combined with the discontinuous $P_2$ 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 $P_4$ bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory.<|reference_end|>
@article{xu2024a,
    title={A nonconforming P3 and discontinuous P2 mixed finite element on
  tetrahedral grids},
    author={Xuejun Xu and Shangyou Zhang},
    journal={arXiv preprint arXiv:2408.10226},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.10226},
    primaryClass={math.NA cs.NA}
}