Convergence analysis of incomplete biquadratic rectangular element for fourth-order singular perturbation problem on anisotropic meshes. (English) Zbl 1470.74064

Summary: The convergence analysis of a Morley type rectangular element for the fourth-order elliptic singular perturbation problem is considered. A counterexample is provided to show that the element is not uniformly convergent with respect to the perturbation parameter. A modified finite element approximation scheme is used to get convergent results; the corresponding error estimate is presented under anisotropic meshes. Numerical experiments are also carried out to demonstrate the theoretical analysis.


74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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[1] Chen, S. C.; Zhao, Y. C.; Shi, D. Y., Non \(C^0\) nonconforming elements for elliptic fourth order singular perturbation problem, Journal of Computational Mathematics, 23, 2, 185-198, (2005) · Zbl 1069.65125
[2] Wang, M.; Xu, J. C.; Hu, Y. C., Modified Morley element method for a fourth order elliptic singular perturbation problem, Journal of Computational Mathematics, 24, 2, 113-120, (2006) · Zbl 1102.65118
[3] Nilssen, T. K.; Tai, X. C.; Winther, R., A robust nonconforming \(H^2\)-element, Mathematics of Computation, 70, 234, 489-505, (2001) · Zbl 0965.65127
[4] Zhang, S.; Wang, M., A posteriori estimator of nonconforming finite element method for fourth order elliptic perturbation problems, Journal of Computational Mathematics, 26, 4, 554-577, (2008) · Zbl 1174.65051
[5] Semper, B., Conforming finite element approximations for a fourth-order singular perturbation problem, SIAM Journal on Numerical Analysis, 29, 4, 1043-1058, (1992) · Zbl 0755.65106
[6] Nitsche, J. A.; Schatz, A. H., Interior estimates for Ritz-Galerkin methods, Mathematics of Computation, 28, 937-958, (1974) · Zbl 0298.65071
[7] Schatz, A. H.; Wahlbin, L. B., On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Mathematics of Computation, 40, 161, 47-89, (1983) · Zbl 0518.65080
[8] Strang, G.; Aziz, A. R., Variational crimes in the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, 689-710, (1972), New York, NY, USA: Academic Press, New York, NY, USA
[9] Stummel, F., The generalized patch test, SIAM Journal on Numerical Analysis, 16, 3, 449-471, (1979) · Zbl 0418.65058
[10] Shi, Z. C., The F-E-M test for convergence of nonconforming finite elements, Mathematics of Computation, 49, 180, 391-405, (1987) · Zbl 0648.65075
[11] Chen, S. C.; Shi, D. Y.; Ichiro, H., Generalized estimate formulation of nonconforming finite elements, Mathematica Numerica Sinica, 22, 3, 295-300, (2000)
[12] Shi, Z. C., On the convergence of the incomplete biquadratic nonconforming plate element, Mathematica Numerica Sinica, 8, 1, 53-62, (1986) · Zbl 0587.73114
[13] Shi, Z. C., The generalized patch test for Zienkiewicz’s triangles, Journal of Computational Mathematics, 2, 3, 279-286, (1984) · Zbl 0564.73069
[14] Shi, Z. C.; Chen, S. C., An analysis of a nine-parameter plate element of Specht, Mathematica Numerica Sinica, 11, 3, 312-318, (1989) · Zbl 0724.73240
[15] Shi, D. Y.; Mao, S. P.; Chen, S. C., On the anisotropic accuracy analysis of ACM’s nonconforming finite element, Journal of Computational Mathematics, 23, 6, 635-646, (2005) · Zbl 1087.65103
[16] Mao, S. P.; Chen, S. C., Convergence analysis of Morley element on anisotropic meshes, Journal of Computational Mathematics, 24, 2, 169-180, (2006) · Zbl 1100.74056
[17] Shi, Z. C.; Chen, Q. Y., A new high accuracy rectangular element, Scince in China A, 30, 6, 504-515, (2000)
[18] Morley, L. S. D., The triangular equilibrium element in the solution of plate bending problems, Aero Quartet, 19, 149-169, (1968)
[19] Lascaux, P.; Lesaint, P., Some nonconforming finite elements for the plate bending problem, Analyse Numérique, 9, 1, 9-53, (1975) · Zbl 0319.73042
[20] Shi, Z. C., Error estimates for the Morley element, Mathematica Numerica Sinica, 12, 2, 113-118, (1990) · Zbl 0850.73337
[21] Mao, S. P.; Shi, Z. C., High accuracy analysis of two nonconforming plate elements, Numerische Mathematik, 111, 3, 407-443, (2009) · Zbl 1155.74045
[22] Wang, M., On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements, SIAM Journal on Numerical Analysis, 39, 2, 363-384, (2001) · Zbl 1069.65116
[23] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, (1978), Amsterdam, The Netherlands: North-Holland, Amsterdam, The Netherlands · Zbl 0383.65058
[24] Zhang, H. Q.; Wang, M., The Mathematical Theory of Finite Elements, (1991), Beijing, China: Science Press, Beijing, China
[25] Apel, T.; Dobrowolski, M., Anisotropic interpolation with applications to the finite element method, Computing, 47, 3-4, 277-293, (1992) · Zbl 0746.65077
[26] Chen, S. C.; Shi, D. Y.; Zhao, Y. C., Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes, IMA Journal of Numerical Analysis, 24, 1, 77-95, (2004) · Zbl 1049.65129
[27] Xie, P. L., Nonconforming finite element methods for fourth order elliptic problems [Ph.D. thesis], (2009), Shanghai, China: Fudan University, Shanghai, China
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