A nonconforming P2 and discontinuous P1 mixed finite element on tetrahedral grids

TL;DR

This paper introduces a new nonconforming P2 finite element combined with a discontinuous P1 element on tetrahedral grids, achieving stable and optimal solutions for the stationary Stokes equations, verified by numerical tests.

Contribution

It develops a novel nonconforming P2 element with bubble functions and demonstrates its stability and optimal convergence when combined with a discontinuous P1 element for Stokes problems.

Findings

Achieves inf-sup stability on tetrahedral grids.

Provides optimal-order convergence for stationary Stokes equations.

Numerical tests confirm theoretical results.

Abstract

A nonconforming $P_2$ finite element is constructed by enriching the conforming $P_2$ finite element space with seven $P_2$ nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming $P_2$ finite element, combined with the discontinuous $P_1$ finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently such a mixed finite element method produces optimal-order convergen solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.