A C0-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1981-2001.

In this paper, a C 0 nonconforming quadrilateral element is proposed to solve the fourth-order elliptic singular perturbation problem. For each convex quadrilateral Q , the shape function space is the union of S 2 1 ( Q * ) and a bubble space. The degrees of freedom are defined by the values at vertices and midpoints on the edges, and the mean values of integrals of normal derivatives over edges. The local basis functions of our element can be expressed explicitly by a new reference quadrilateral rather than by solving a linear system. It is shown that the method converges uniformly in the perturbation parameter. Lastly, numerical tests verify the convergence analysis.

DOI : 10.1051/m2an/2018033
Classification : 65N30
Mots-clés : Singular perturbation problem, quadrilateral element, uniformly convergent
Bao, Yuan 1 ; Meng, Zhaoliang 1 ; Luo, Zhongxuan 1

1 
@article{M2AN_2018__52_5_1981_0,
     author = {Bao, Yuan and Meng, Zhaoliang and Luo, Zhongxuan},
     title = {A {C\protect\textsuperscript{0}-nonconforming} quadrilateral finite element for the fourth-order elliptic singular perturbation problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1981--2001},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {5},
     year = {2018},
     doi = {10.1051/m2an/2018033},
     zbl = {1426.65169},
     mrnumber = {3885703},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2018033/}
}
TY  - JOUR
AU  - Bao, Yuan
AU  - Meng, Zhaoliang
AU  - Luo, Zhongxuan
TI  - A C0-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 1981
EP  - 2001
VL  - 52
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2018033/
DO  - 10.1051/m2an/2018033
LA  - en
ID  - M2AN_2018__52_5_1981_0
ER  - 
%0 Journal Article
%A Bao, Yuan
%A Meng, Zhaoliang
%A Luo, Zhongxuan
%T A C0-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 1981-2001
%V 52
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2018033/
%R 10.1051/m2an/2018033
%G en
%F M2AN_2018__52_5_1981_0
Bao, Yuan; Meng, Zhaoliang; Luo, Zhongxuan. A C0-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1981-2001. doi : 10.1051/m2an/2018033. http://www.numdam.org/articles/10.1051/m2an/2018033/

[1] A. Adini and R.W. Clough, Analysis of plate bending by the finite element method, Technical report, NSF Report for Grant G. 7337 (1960).

[2] G. Awanou, Robustness of a spline element method with constraints. J. Sci. Comput. 36 (2008) 421–432. | DOI | MR | Zbl

[3] I. Babuška, The finite element method for elliptic equations with discontinuous coefficients. Computing 5 (1970) 207–213. | DOI | MR | Zbl

[4] G.P. Bazeley, Y.K. Cheung, B.M. Irons and O.C. Zienkiewicz, Triangular elements in plate bending-conforming and nonconforming solutions, in Proc. Conf. Matrix Methods in Struct. Mech., AirForce Inst. of Tech., Wright Patterson AF Base, Ohio (1965).

[5] F.K. Bogner, The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulae, in Proc. Conf. Matrix Methods in Struct. Mech., AirForce Inst. of Tech., Wright Patterson AF Base, Ohio (1965).

[6] S.C. Brenner and M. Neilan, A C0 interior penalty method for a fourth order elliptic singular perturbation problem. SIAM J. Numer. Anal. 49 (2011) 869–892. | DOI | MR | Zbl

[7] S. Chen, M. Liu and Z. Qiao, An anisotropic nonconforming element for fourth order elliptic singular perturbation problem. Int. J. Numer. Anal. Model. 7 (2010) 766–784. | MR | Zbl

[8] S. Chen and Z. Shi, Double set parameter method of constructing stiffness matrices. Numer. Math. Sinica 13 (1991) 286–296. | MR | Zbl

[9] S. Chen, Y. Zhao and D. Shi, Non C0 nonconforming elements for elliptic fourth order singular perturbation problem. J. Comput. Math. 23 (2005) 185–198. | MR | Zbl

[10] B.F. De  Veubeke, A conforming finite element for plate bending. Int. J. Solids Struct. 4 (1968) 95–108 | DOI | Zbl

[11] G. Farin, Triangular Bernstein-Bézier patches. Comput. Aided Geom. Design 3 (1986) 83–127. | DOI | MR

[12] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Publishing, Inc., Boston (1985). | MR | Zbl

[13] H. Guo, C. Huang and Z. Zhang, Superconvergence of conforming finite element for fourth-order singularly perturbed problems of reaction diffusion type in 1D. Numer. Methods Part. Differ. Equ. 30 (2014) 550–566. | DOI | MR | Zbl

[14] J. Guzmán, D. Leykekhman and M. Neilan, A family of non-conforming elements and the analysis of Nitsche’s method for a singularly perturbed fourth order problem. Calcolo 49 (2012) 95–125. | DOI | MR | Zbl

[15] J. Hu, Y. Huang and S. Zhang, The lowest order differentiable finite element on rectangular grids. SIAM J. Numer. Anal. 49 (2011) 1350–1368. | DOI | MR | Zbl

[16] I. Kim, Z. Luo, Z. Meng, H. Nam, C. Park and D. Sheen, A piecewise P2-nonconforming quadrilateral finite element. ESAIM: M2AN 47 (2013) 689–715. | DOI | Numdam | MR | Zbl

[17] C. Li and R. Wang, A new 8-node quadrilateral spline finite element. J. Comput. Appl. Math. 195 (2006) 54–65. | DOI | MR | Zbl

[18] Z. Meng, J. Cui and Z. Luo, A new rotated nonconforming quadrilateral element. J. Sci. Comput. 74 (2018) 324–335. | DOI | MR | Zbl

[19] L.S.D. Morley, The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19 (1968) 149–169. | DOI

[20] T.K. Nilssen, X. Tai and R. Winther, A robust nonconforming H2-element. Math. Comput. 70 (2001) 489–505. | DOI | MR | Zbl

[21] C. Park and D. Sheen, A quadrilateral Morley element for biharmonic equations. Numer. Math. 124 (2013) 395–413. | DOI | MR | Zbl

[22] H.-G. Roos and M. Stynes, A uniformly convergent discretization method for a fourth order singular perturbation problem. Bonn. Math. Schr. 228 (1991) 30–40. | MR | Zbl

[23] B. Semper, Conforming finite element approximations for a fourth-order singular perturbation problem. SIAM J. Numer. Anal. 29 (1992) 1043–1058. | DOI | MR | Zbl

[24] D. Shi and P. Xie, A robust double set parameter nonconforming rectangular element for fourth order elliptic singular perturbation problems. Proc. Environ. Sci. 10 (2011) 854–868. | DOI

[25] Z. Shi, On the convergence of the incomplete biquadratic nonconforming plate element. Math. Numer. Sinica 8 (1986) 53–62. | MR | Zbl

[26] Z. Shi, On the error estimates of Morley element. Numer. Math. 12 (1990) 113–118. | MR | Zbl

[27] L. Wang, Y. Wu and X. Xie, Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems. Numer. Method. Part. Differ. Equ. 29 (2013) 721–737. | DOI | MR | Zbl

[28] M. Wang, J. Xu and Y. Hu, Modified Morley element method for a fourth order elliptic singular perturbation problem. J. Comput. Math. 24 (2006) 113–120. | MR | Zbl

[29] P. Xie and M. Hu, Convergence analysis of incomplete biquadratic rectangular element for fourth-order singular perturbation problem on anisotropic meshes. Abstr. Appl. Anal. 2014 (2014) 1–11. | DOI | MR | Zbl

[30] P. Xie, D. Shi and H. Li, A new robust C0-type nonconforming triangular element for singular perturbation problems. Appl. Math. Comput. 217 (2010) 3832–3843. | DOI | MR | Zbl

Cité par Sources :