## A piecewise *P*_{2}-nonconforming
quadrilateral finite element

^{1} Department of Mathematics, Seoul
National University, 151-747
Seoul,
Korea

ikim@snu.ac.kr; sheen@snu.ac.kr
^{2} School of Mathematical Sciences,
Dalian University of Technology, Dalian, China

zxluo@dlut.edu.cn; mzhl@dlut.edu.cn
^{3} Interdisciplinary Program in
Computational Sciences and Technology, Seoul National University,
151-747
Seoul,
Korea

dongwoosheen@gmail.com; lamyun96@snu.ac.kr
^{4} Department of Mathematics, Konkuk
University, 143-701
Seoul,
Korea

cpark@konkuk.ac.kr

Received:
3
January
2012

Revised:
16
July
2012

We introduce a piecewise *P*_{2}-nonconforming quadrilateral
finite element. First, we decompose a convex quadrilateral into the union of four
triangles divided by its diagonals. Then the finite element space is defined by the set of
all piecewise *P*_{2}-polynomials that are quadratic in each
triangle and continuously differentiable on the quadrilateral. The degrees of freedom
(DOFs) are defined by the eight values at the two Gauss points on each of the four edges
plus the value at the intersection of the diagonals. Due to the existence of one linear
relation among the above DOFs, it turns out the DOFs are eight. Global basis functions are
defined in three types: vertex-wise, edge-wise, and element-wise types. The corresponding
dimensions are counted for both Dirichlet and Neumann types of elliptic problems. For
second-order elliptic problems and the Stokes problem, the local and global interpolation
operators are defined. Also error estimates of optimal order are given in both broken
energy and *L*^{2}(*Ω*) norms. The proposed element
is also suitable to solve Stokes equations. The element is applied to approximate each
component of velocity fields while the discontinuous
*P*_{1}-nonconforming quadrilateral element is adopted to
approximate the pressure. An optimal error estimate in energy norm is derived. Numerical
results are shown to confirm the optimality of the presented piecewise
*P*_{2}-nonconforming element on quadrilaterals.

Mathematics Subject Classification: 65N30 / 76M10

Key words: Nonconforming finite element / Stokes problem / elliptic problem / quadrilateral

*© EDP Sciences, SMAI, 2013*