A Nonconforming P2 and Discontinuous P1 Mixed Finite Element on Tetrahedral Grids

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 convergent solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.