## 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.

### MSC:

 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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### References:

 [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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