A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.10226
- arXiv:
- arXiv:2408.10226
- Bibcode:
- 2024arXiv240810226X
- Keywords:
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- Mathematics - Numerical Analysis