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/

 R. Altmann and C. Carstensen, p1-nonconforming finite elements on triangulations into triangles and quadrilaterals. SIAM J. Numer. Anal. 50 (2012) 418-438. | MR 2914269 | Zbl 1251.65156

 D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337-344. | MR 799997 | Zbl 0593.76039

 D. N. Arnold and R. Winther, Nonconforming mixed elements for elasticity. Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday. Math. Models Methods Appl. Sci. 13 (2003) 295-307. | MR 1977627 | Zbl 1057.74036

 I. Babuška and M. Suri, Locking effect in the finite element approximation of elasticity problem. Numer. Math. 62 (1992) 439-463. | MR 1174468 | Zbl 0762.65057

 I. Babuška and M. Suri, On locking and robustness in the finie element method. SIAM J. Numer. Anal. 29 (1992) 1261-1293. | MR 1182731 | Zbl 0763.65085

 R. Bank and B. Welfert, A comparison between the mini-element and the Petrov-Galerkin formulations for the generalized Stokes problem. Comput. Methods Appl. Mech. Eng. 83 (1990) 61-68. | MR 1078695 | Zbl 0732.65100

 J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 113-124. | MR 263214 | Zbl 0201.07803

 S. Brenner and L. Scott, The Mathematical Theorey of Finite Element Methods. Springer-Verlag, New York (1994). | MR 1278258 | Zbl 1135.65042

 S.C. Brenner and L.Y. Sung, Linear finite element methods for planar elasticity. Math. Comput. 59 (1992) 321-338. | Zbl 0766.73060

 F. Brezzi, M.-O. Bristeau, L.P. Franca, M. Mallet and G. Rogé, A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Meth. Appl. Mech. Eng. 96 (1992) 117-129. | MR 1159592 | Zbl 0756.76044

 F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems. ESAIM-Math. Model. Numer. Anal. 43 (2009) 277-295. | Numdam | MR 2512497 | Zbl 1177.65164

 F. Brezzi and J. Douglas, Jr. Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225-236. | MR 946377 | Zbl 0669.76052

 F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York. Springer Series Comput. Math. 15 (1991). | MR 1115205 | Zbl 0788.73002

 F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. (2006) 1872-1896. | MR 2192322 | Zbl 1108.65102

 A.N. Brooks and T.J.R. Hughes, Streamline upwind Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32 (1982) 199-259. | MR 679322 | Zbl 0497.76041

 Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming quadrilateral finite elements: A correction. Calcolo 37 (2000) 253-254. | MR 1812789 | Zbl 1012.65124

 Z. Cai, J. Douglas, Jr. and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36 (1999) 215-232. | MR 1740354 | Zbl 0947.76047

 C. Carstensen and J. Hu, A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107 (2007) 473-502. | MR 2336116 | Zbl 1127.65083

 P.G. Ciarlet, The Finite Element Method for Elliptic Equations. North-Holland, Amsterdam (1978). | MR 520174

 G.R. Cowper, Gaussian quadrature formulas for triangles. Int. J. Num. Meth. Eng. 7 (1973) 405-408. | Zbl 0265.65013

 M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Math. Model. Anal. Numer. R-3 (1973) 33-75. | Numdam | MR 343661 | Zbl 0302.65087

 L.B. Da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comp. Phys. 228 (2009) 7215-7232. | MR 2568590 | Zbl 1172.76032

 L.B. Da Veiga, K. Lipnikov and G. Manzini, Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113 (2009) 325-356. | MR 2534128 | Zbl 1183.65132

 L.B. Da Veiga and G. Manzini, A higher-order formulation of the mimetic finite difference method. SIAM J. Sci. Comput. 31 (2008) 732-760. | MR 2460797 | Zbl 1185.65201

 J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. ESAIM Math. Model. Numer. Anal. 33 (1999) 747-770. | Numdam | MR 1726483 | Zbl 0941.65115

 J. Douglas, Jr. and J. Wang. An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52 (1989) 495-508. | MR 958871 | Zbl 0669.76051

 R.S. Falk, Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57 (1991) 529-550. | MR 1094947 | Zbl 0747.73044

 M. Farhloul and M. Fortin, A mixed nonconforming finite element for the elasticity and Stokes problems. Math. Models Methods Appl. Sci. 9 (1999) 1179-1199. | MR 1722052 | Zbl 1044.74042

 M. Fortin, A three-dimensional quadratic nonconforming element. Numer. Math. 46 (1985) 269-279. | MR 787211 | Zbl 0577.65008

 M. Fortin and M. Soulie, A non-conforming piecewise quadratic finite element on the triangle. Int. J. Numer. Meth. Eng. 19 (1983) 505-520. | MR 702056 | Zbl 0514.73068

 L. Franca, S. Frey and T. Hughes, Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Eng. 95 (1992) 221-242. | MR 1155924 | Zbl 0759.76040

 V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin (1986). | MR 851383 | Zbl 0585.65077

 V. Gyrya and K. Lipnikov, High-order mimetic finite difference method for diffusion problems on polygonal meshes. J. Comput. Phys. 227 (2008) 8841-8854. | MR 2459538 | Zbl 1152.65101

 H. Han, Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (1984) 223-233. | MR 815417 | Zbl 0573.65083

 P. Hood and C. Taylor, A numerical solution for the Navier-Stokes equations using the finite element technique. Computers Fluids 1 (1973) 73-100. | MR 339677 | Zbl 0328.76020

 T.J.R. Hughes and A.N. Brooks, A multidimensional upwind scheme with no crosswind diffusion, in Finite Element Methods for Convection Dominated Flows, edited by T.J.R. Hughes. ASME, New York (1979) 19-35. | MR 571681 | Zbl 0423.76067

 B.M. Irons and A. Razzaque, Experience with the patch test for convergence of finite elements, in The Mathematics of Foundation of the Finite Element Methods with Applications to Partial Differential Equations, edited by A.K. Aziz. Academic Press, New York (1972) 557-587. | MR 423839 | Zbl 0279.65087

 P. Klouček, B. Li and M. Luskin, Analysis of a class of nonconforming finite elements for crystalline microstructures. Math. Comput. 65 (1996) 1111-1135. | MR 1344616 | Zbl 0903.65081

 M. Köster, A. Quazzi, F. Schieweck, S. Turek and P. Zajac, New robust nonconforming finite elements of higher order. Appl. Numer. Math. 62 (2012) 166-184. | MR 2878019 | Zbl 1238.65112

 C.-O. Lee, J. Lee and D. Sheen, A locking-free nonconforming finite element method for planar elasticity. Adv. Comput. Math. 19 (2003) 277-291. | MR 1973469 | Zbl 1064.74165

 H. Lee and D. Sheen, A new quadratic nonconforming finite element on rectangles. Numer. Methods Partial Differ. Equ. 22 (2006) 954-970. | MR 2230281 | Zbl 1097.74059

 P. Lesaint, On the convergence of Wilson's nonconforming element for solving the elastic problem. Comput. Methods Appl. Mech. Eng. 7 (1976) 1-76. | MR 455479 | Zbl 0345.65058

 B. Li and M. Luskin, Nonconforming finite element approximation of crystalline microstructure. Math. Comput. 67 (1998) 917-946. | MR 1459391 | Zbl 0901.73076

 Z.X. Luo, Z.L. Meng and C.M. Liu, Computational Geometry - Theory and Applications of Surface Representation. Sinica Academic Press, Beijing (2010). | Zbl 0679.68206

 P. Ming and Z.-C. Shi, Nonconforming rotated Q1 element for Reissner-Mindlin plate. Math. Models Methods Appl. Sci. 11 (2001) 1311-1342. | MR 1859825 | Zbl 1037.74048

 C. Park and D. Sheen. P1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 624-640. | MR 2004191 | Zbl 1048.65114

 R. Pierre, Simple C0 approximations for the computation of incompressible flows. Comput. Methods Appl. Mech. Eng. 68 (1988) 205-227. | MR 942313 | Zbl 0628.76040

 R. Pierre, Regularization procedures of mixed finite element approximations of the Stokes problem. Numer. Methods Partial Differ. Equ. 5 (1989) 241-258. | MR 1107887 | Zbl 0672.76038

 R. Rannacher and S. Turek. Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8 (1992) 97-111. | MR 1148797 | Zbl 0742.76051

 G. Sander and P. Beckers, The influence of the choice of connectors in the finite element method. Int. J. Numer. Methods Eng. 11 (1977) 1491-1505. | MR 502734 | Zbl 0439.65090

 Z.-C. Shi, A convergence condition for the quadrilateral Wilson element. Numer. Math. 44 (1984) 349-361. | MR 757491 | Zbl 0581.65008

 Z.-C. Shi, On the convergence properties of the quadrilateral elements of Sander and Beckers. Math. Comput. 42 (1984) 493-504. | MR 736448 | Zbl 0557.65072

 G. Strang, Variational crimes in the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, edited by A.K. Aziz. New York, Academic Press (1972) 689-710. | MR 413554 | Zbl 0264.65068

 G. Strang and G.J. Fix, An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973). | MR 443377 | Zbl 0356.65096

 R. Wang, Multivariate Spline Functions and Their Applications. Science Press, Kluwer Academic Publishers (1994). | Zbl 1002.41001

 E. L. Wilson, R. L. Taylor, W. P. Doherty and J. Ghaboussi, Incompatible displacement models, in Numerical and Computer Method in Structural Mechanics, Academic Press, New York (1973) 43-57.

 Z. Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal. 34 (1997) 640-663. | MR 1442932 | Zbl 0870.73074