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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 99
PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping
Paper 256

On the Convergence of a Nonconforming Plane Quadrilateral Finite Element

M. Saje and R. Flajs

Faculty of Civil and Geodetic Engineering, University of Ljubljana, Slovenia

Full Bibliographic Reference for this paper
M. Saje, R. Flajs, "On the Convergence of a Nonconforming Plane Quadrilateral Finite Element", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 256, 2012. doi:10.4203/ccp.99.256
Keywords: nonconforming plane quadrilateral finite element, convergence, generalized patch test.

Summary
In this paper the sufficient conditions for convergence of the quadrilateral nonconforming finite element RQ6, are derived. This element was recently introduced by Cheung et al. [1] when solving the Dirichlet boundary value problem of a second order elliptic equation on a convex domain. The analytical proof of convergence for this kind of finite element is based on Stummel's generalized patch test [2], the approximating condition and the degrees of freedom unisolvence requirement. In deriving the error estimates, the methodology of Shi [3] was partially followed employing some inequalities given by Brenner and Scott [4].

As a result of the strict use of the Cartesian base functions, the interpolation matrix of the finite element may experience singularity or rank deficiency which results in finite element degrees of freedom non-unisolvence (termed here the 'finite element nonunisolvence' with a 'nonunisolvent quadrilateral' domain). This essential drawback is here resolved by a unique further division of such a quadrilateral into more quadrilaterals with a briefly described procedure named quadriangulation, practically performed by a sequence of properly chosen triangulations.

The nonconforming finite element passes the Irons' patch test, which is simple to employ. However, according to Stummel, Irons' patch test as the convergence condition is in general neither sufficient nor necessary for convergence of a nonconforming finite element, and is thus of a limited applicability. Because the mathematics involved in proving convergence analytically employing the Stummel generalised patch test is highly specific and demanding, analytical proofs of convergence of nonconforming finite elements have been performed only occasionally for a specific nonconforming finite element [3]. The presented proof could thus adds some light in this area.

The derivations show that the RQ6 finite element is convergent even if its quadrilateral domain is nonconvex which serves as a theoretically interesting advantage when compared to standard isoparametric elements. However from the numerical results obtained it is clearly seen that the results obtained using dividing schemes with only convex quadrilaterals are somewhat more accurate compared to nonconvex quadrilaterals.

The results of various numerical examples completely confirm the present theoretical findings, in particular the linear rate of convergence in the energy norm, and the quadratic convergence in the L2 norm for both convex and nonconvex meshes.

References
1
Y.K. Cheung, Y.X. Zhang, W.J. Chen, "A refined nonconforming plane quadrilateral element", Comput.Struct., 78, 669-709, 2000. doi:10.1016/S0045-7949(00)00049-3
2
F. Stummel, "The generalised patch test", SIAM J. Numer. Anal., 16(3), 449-471, 1979.
3
Z.C. Shi, "A convergence condition for the quadrilateral Wilson element", Numer. Math., 44, 349-361, 1984. doi:10.1007/BF01405567
4
S.C. Brenner, L.R. Scott, "The mathematical theory of finite element methods", 2nd ed., Springer, New York, 2002. doi:10.1007/978-0-387-75934-0

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