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September  2017, 22(7): 2831-2856. doi: 10.3934/dcdsb.2017153

## New convergence analysis for assumed stress hybrid quadrilateral finite element method

 School of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author: Xiaoping Xie, xpxie@scu.edu.cn

Received  June 2016 Revised  March 2017 Published  May 2017

Fund Project: This work was supported by Major Research Plan of National Natural Science Foundation of China (91430105) and National Natural Science Foundation of China (11171239,11401407)

New error estimates are established for Pian and Sumihara's (PS) 4-node assumed stress hybrid quadrilateral element [T.H.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg., 20 (1984), 1685-1695], which is widely used in engineering computation. Based on an equivalent displacement-based formulation to the PS element, we show that the numerical strain and a postprocessed numerical stress are uniformly convergent with respect to the Lamé constant $λ$ on the meshes produced through the uniform bisection procedure. Within this analysis framework, we also show that both the numerical strain and stress are uniformly convergent on meshes which are stable for the $Q_1-P_0$ Stokes element.

Citation: Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153
##### References:
 [1] D. Boffi, F. Brezzi and M. Fortin, Finite elements for the Stokes problem, in Mixed Finite Elements, Compatibility Conditions, and Applications (eds. D. Boffi and L. Gastaldi), Springer Berlin-Heidelberg, (2008), 45-100. Google Scholar [2] D. Braess, Enhanced assumed strain elements and locking in membrane problems, Computer Methods in Applied Mechanics and Engineering, 165 (1998), 155-174. doi: 10.1016/S0045-7825(98)00037-1. Google Scholar [3] D. Braess, C. Carstensen and B. D. Reddy, Uniform convergence and a posteriori error estimators for the enhanced strain finite element method, Numerische Mathematik, 96 (2004), 461-479. doi: 10.1007/s00211-003-0486-5. Google Scholar [4] S. C. Brenner and L. Y. Sung, Linear finite element methods for planar linear elasticity, Mathematics of Computation, 59 (1992), 321-338. doi: 10.2307/2153060. Google Scholar [5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3$^{rd}$ edition, Springer-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0. Google Scholar [6] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics Springer-Verlag, Berlin, 1976. Google Scholar [7] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, New York, 1986. doi: 10.1007/978-3-642-61623-5. Google Scholar [8] T. H. H. Pian, Derivation of element stiffness matrices by assumed stress distributions, AIAA Journal, 7 (1964), 1333-1336. Google Scholar [9] T. H. H. Pian and K. Sumihara, Rational approach for assumed stress finite elements, International Journal for Numerical Methods in Engineering, 20 (1984), 1685-1695. Google Scholar [10] J. Pitkäranta and R. Stenberg, Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes, in The Mathematics of Finite Elements and Applications V, (de. Whiteman J), Academic Press, (1985), 325-334. Google Scholar [11] J. Qin and S. Zhang, On the selective local stabilization of the mixed Q1-P0 element, International Journal for Numerical Methods in Fluids, 55 (2007), 1121-1141. doi: 10.1002/fld.1505. Google Scholar [12] B. D. Reddy and J. C. Simo, Stability and convergence of a class of enhanced strain methods, SIAM Journal on Numerical Analysis, 32 (1995), 1705-1728. doi: 10.1137/0732077. Google Scholar [13] Z. Shi, A convergence condition for the quadrilateral wilson element, Numerische Mathematik, 44 (1984), 349-361. doi: 10.1007/BF01405567. Google Scholar [14] J. C. Simo and M. S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering, 29 (1990), 1595-1638. doi: 10.1002/nme.1620290802. Google Scholar [15] R. L. Taylor, E. L. Wilson and P. J. Beresford, A nonconforming element for stress analysis, International Journal for Numerical Methods in Engineering, 10 (1976), 1211-1219. Google Scholar [16] E. L. Wilson, R. L. Taylor, W. P. Doherty and J. Ghaboussi, Incompatible displacement models, in Numerical Computer Methods in Structural Mechanics, Academic Press, (1973), 43-57. Google Scholar [17] X. Xie and T. Zhou, Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals, International Journal for Numerical Methods in Engineering, 59 (2004), 293-313. doi: 10.1002/nme.877. Google Scholar [18] G. Yu, X. Xie and C. Carstensen, Uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 2421-2433. doi: 10.1016/j.cma.2011.03.018. Google Scholar [19] Z. Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM Journal on Numerical Analysis, 34 (1997), 640-663. doi: 10.1137/S0036142995282492. Google Scholar [20] T. Zhou and X. Xie, A unified analysis for stress/strain hybrid methods of high performance, Computer Methods in Applied Mechanics and Engineering, 191 (2002), 4619-4640. doi: 10.1016/S0045-7825(02)00396-1. Google Scholar

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##### References:
 [1] D. Boffi, F. Brezzi and M. Fortin, Finite elements for the Stokes problem, in Mixed Finite Elements, Compatibility Conditions, and Applications (eds. D. Boffi and L. Gastaldi), Springer Berlin-Heidelberg, (2008), 45-100. Google Scholar [2] D. Braess, Enhanced assumed strain elements and locking in membrane problems, Computer Methods in Applied Mechanics and Engineering, 165 (1998), 155-174. doi: 10.1016/S0045-7825(98)00037-1. Google Scholar [3] D. Braess, C. Carstensen and B. D. Reddy, Uniform convergence and a posteriori error estimators for the enhanced strain finite element method, Numerische Mathematik, 96 (2004), 461-479. doi: 10.1007/s00211-003-0486-5. Google Scholar [4] S. C. Brenner and L. Y. Sung, Linear finite element methods for planar linear elasticity, Mathematics of Computation, 59 (1992), 321-338. doi: 10.2307/2153060. Google Scholar [5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3$^{rd}$ edition, Springer-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0. Google Scholar [6] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics Springer-Verlag, Berlin, 1976. Google Scholar [7] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag, New York, 1986. doi: 10.1007/978-3-642-61623-5. Google Scholar [8] T. H. H. Pian, Derivation of element stiffness matrices by assumed stress distributions, AIAA Journal, 7 (1964), 1333-1336. Google Scholar [9] T. H. H. Pian and K. Sumihara, Rational approach for assumed stress finite elements, International Journal for Numerical Methods in Engineering, 20 (1984), 1685-1695. Google Scholar [10] J. Pitkäranta and R. Stenberg, Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes, in The Mathematics of Finite Elements and Applications V, (de. Whiteman J), Academic Press, (1985), 325-334. Google Scholar [11] J. Qin and S. Zhang, On the selective local stabilization of the mixed Q1-P0 element, International Journal for Numerical Methods in Fluids, 55 (2007), 1121-1141. doi: 10.1002/fld.1505. Google Scholar [12] B. D. Reddy and J. C. Simo, Stability and convergence of a class of enhanced strain methods, SIAM Journal on Numerical Analysis, 32 (1995), 1705-1728. doi: 10.1137/0732077. Google Scholar [13] Z. Shi, A convergence condition for the quadrilateral wilson element, Numerische Mathematik, 44 (1984), 349-361. doi: 10.1007/BF01405567. Google Scholar [14] J. C. Simo and M. S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering, 29 (1990), 1595-1638. doi: 10.1002/nme.1620290802. Google Scholar [15] R. L. Taylor, E. L. Wilson and P. J. Beresford, A nonconforming element for stress analysis, International Journal for Numerical Methods in Engineering, 10 (1976), 1211-1219. Google Scholar [16] E. L. Wilson, R. L. Taylor, W. P. Doherty and J. Ghaboussi, Incompatible displacement models, in Numerical Computer Methods in Structural Mechanics, Academic Press, (1973), 43-57. Google Scholar [17] X. Xie and T. Zhou, Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals, International Journal for Numerical Methods in Engineering, 59 (2004), 293-313. doi: 10.1002/nme.877. Google Scholar [18] G. Yu, X. Xie and C. Carstensen, Uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 2421-2433. doi: 10.1016/j.cma.2011.03.018. Google Scholar [19] Z. Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM Journal on Numerical Analysis, 34 (1997), 640-663. doi: 10.1137/S0036142995282492. Google Scholar [20] T. Zhou and X. Xie, A unified analysis for stress/strain hybrid methods of high performance, Computer Methods in Applied Mechanics and Engineering, 191 (2002), 4619-4640. doi: 10.1016/S0045-7825(02)00396-1. Google Scholar
A stable mesh for $Q_1\!-\!P_0$ Stokes element
$\widehat\chi\circ F_K^{-1}$ for some $K\in\mathcal T_{2h}$
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