A piecewise P₂-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

Abstract

We introduce a piecewise P₂-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 P₂-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 L²(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 P₁-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 P₂-nonconforming element on quadrilaterals.