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 vertex 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 + z2)],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 +yz2,z2x + zx2}, respectively, while the sixth type adds the span of {xyz,xy2z2,x2yz2,x2y2zx}. We compare these three elements with the Meng-Sheen-Luo-Kim element numerically and give rates of convergence for Poisson equations.