NUMERICAL STUDY ON THREE-DIMENSIONAL QUADRATIC NONCONFORMING BRICK ELEMENTS

Abstract

Recently a new nonconforming brick element of fourteen DOFs with

quadratic convergence for the energy norm is introduced by Meng, Sheen,

Luo, and Kim [23]. The purpose of this paper is to compare this element

with the brick elements introduced by Smith and Kidger [31]. The above

elements have fourteen degrees of freedom which contain the eight ver

tex values and the six barycenter values at surfaces. The underlying

element are based on P2. The finite element of Meng-Sheen-Luo-Kim

adds the span of four polynomials {xyz, x[x2 − 3 5

(y2 + z2)], y[y2 − 3 5

(x2 +

z

2

)], z[z2 − 3 5

(x2 + y2)]}, while the Smith-Kidger elements add the span

of four other polynomials. In this paper, we particularly consider the

two classes of Smith-Kidger elements. The first and fifth types add

the span of {xyz, x2y, y2, z2x} and the span of {xyz, x2y + xy2, y2z +

yz

2

, z

2

x + zx2 }, respectively, while the sixth type adds the span of

Meng-Sheen-Luo-Kim element numerically and give rates of convergence {xyz, xy2z2, x2yz2, x2y2zx}. We compare these three elements with the

for Poisson equations