A nonconforming P3+B4 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 nine \(P_4\) 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.