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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.
@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 },
pages = {689--715},
publisher = {EDP-Sciences},
volume = {47},
number = {3},
year = {2013},
doi = {10.1051/m2an/2012044},
zbl = {1270.65067},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012044/}
}
TY - JOUR AU - Kim, Imbunm AU - Luo, Zhongxuan AU - Meng, Zhaoliang AU - NAM, Hyun AU - Park, Chunjae AU - Sheen, Dongwoo TI - A piecewise $P_2$-nonconforming quadrilateral finite element JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 689 EP - 715 VL - 47 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012044/ DO - 10.1051/m2an/2012044 LA - en ID - M2AN_2013__47_3_689_0 ER -
%0 Journal Article %A Kim, Imbunm %A Luo, Zhongxuan %A Meng, Zhaoliang %A NAM, Hyun %A Park, Chunjae %A Sheen, Dongwoo %T A piecewise $P_2$-nonconforming quadrilateral finite element %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 689-715 %V 47 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012044/ %R 10.1051/m2an/2012044 %G en %F 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 , Tome 47 (2013) no. 3, pp. 689-715. doi: 10.1051/m2an/2012044
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