A piecewise ${P}_{2}$-nonconforming quadrilateral finite element
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 3, p. 689-715

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.

DOI : https://doi.org/10.1051/m2an/2012044
Classification:  65N30,  76M10
Keywords: nonconforming finite element, Stokes problem, elliptic problem, quadrilateral
@article{M2AN_2013__47_3_689_0,
author = {Kim, Imbunm and Luo, Zhongxuan and Meng, Zhaoliang and NAM, Hyun and Park, Chunjae and Sheen, Dongwoo},
title = {A piecewise $P\_2$-nonconforming quadrilateral finite element},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {47},
number = {3},
year = {2013},
pages = {689-715},
doi = {10.1051/m2an/2012044},
zbl = {1270.65067},
language = {en},
url = {http://www.numdam.org/item/M2AN_2013__47_3_689_0}
}

Kim, Imbunm; Luo, Zhongxuan; Meng, Zhaoliang; NAM, Hyun; Park, Chunjae; Sheen, Dongwoo. A piecewise $P_2$-nonconforming quadrilateral finite element. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 3, pp. 689-715. doi : 10.1051/m2an/2012044. http://www.numdam.org/item/M2AN_2013__47_3_689_0/

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