A nonconforming P3 and discontinuous P2 mixed finite element on tetrahedral grids

TL;DR

This paper introduces a novel nonconforming P3 finite element combined with a discontinuous P2 element for tetrahedral grids, achieving inf-sup stability and divergence-free velocity solutions for Stokes equations.

Contribution

It develops a new nonconforming P3 finite element with P4 bubbles that, together with a discontinuous P2 element, ensures stability and divergence-free solutions for Stokes problems on tetrahedral meshes.

Findings

The method is inf-sup stable for Stokes equations.

Discrete velocity remains locally divergence-free.

Numerical tests confirm theoretical results.

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.