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A piecewise P2-nonconformingquadrilateral finite element

Published online by Cambridge University Press:  04 March 2013

Imbunm Kim ,
Zhongxuan Luo ,
Zhaoliang Meng ,
Hyun Nam ,
Chunjae Park  and
Dongwoo Sheen
Imbunm Kim
Affiliation:
Department of Mathematics, Seoul National University, 151-747 Seoul, Korea. ikim@snu.ac.kr; sheen@snu.ac.kr
Zhongxuan Luo
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian, China; zxluo@dlut.edu.cn; mzhl@dlut.edu.cn
Zhaoliang Meng
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian, China; zxluo@dlut.edu.cn; mzhl@dlut.edu.cn
Hyun Nam
Affiliation:
Interdisciplinary Program in Computational Sciences and Technology, Seoul National University, 151-747 Seoul, Korea; dongwoosheen@gmail.com; lamyun96@snu.ac.kr
Chunjae Park
Affiliation:
Department of Mathematics, Konkuk University, 143-701 Seoul, Korea; cpark@konkuk.ac.kr
Dongwoo Sheen
Affiliation:
Department of Mathematics, Seoul National University, 151-747 Seoul, Korea. ikim@snu.ac.kr; sheen@snu.ac.kr Interdisciplinary Program in Computational Sciences and Technology, Seoul National University, 151-747 Seoul, Korea; dongwoosheen@gmail.com; lamyun96@snu.ac.kr
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Abstract

We introduce a piecewise P2-nonconforming quadrilateralfinite element. First, we decompose a convex quadrilateral into the union of fourtriangles divided by its diagonals. Then the finite element space is defined by the set ofall piecewise P2-polynomials that are quadratic in eachtriangle 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 edgesplus the value at the intersection of the diagonals. Due to the existence of one linearrelation among the above DOFs, it turns out the DOFs are eight. Global basis functions aredefined in three types: vertex-wise, edge-wise, and element-wise types. The correspondingdimensions are counted for both Dirichlet and Neumann types of elliptic problems. Forsecond-order elliptic problems and the Stokes problem, the local and global interpolationoperators are defined. Also error estimates of optimal order are given in both brokenenergy and L2(Ω) norms. The proposed elementis also suitable to solve Stokes equations. The element is applied to approximate eachcomponent of velocity fields while the discontinuousP1-nonconforming quadrilateral element is adopted toapproximate the pressure. An optimal error estimate in energy norm is derived. Numericalresults are shown to confirm the optimality of the presented piecewiseP2-nonconforming element on quadrilaterals.

Keywords

Nonconforming finite elementStokes problemelliptic problemquadrilateral
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 689 - 715
Copyright
© EDP Sciences, SMAI, 2013

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