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**Analysis of some mixed finite element methods for plane elasticity equations.**
*(English)*
Zbl 0537.73057

Mixed finite element models combined with the penalty method for solving the two-dimensional elasticity equations are analyzed. In the first model, which was proposed recently by R. Taylor and O. C. Zienkiewicz [Energy methods in finite element analysis, Vol. dedic. to the Mem. of B. Fraejis de Veubeke, 153–174 (1979; Zbl 0418.73069)], stresses are approximated by continuous piecewise bilinear functions on a rectangular grid and displacements are taken to be piecewise constant on the same grid. The convergence rate \(O(h^{3/2})\) for stresses in \(L_ 2\) is proved.

Another class of algorithms (method II) is based on the equilibrium composite quadrilateral element considered by B. Fraeijs de Veubeke [in: Stress analysis: Recent developments in numerical and experimental methods, O. C. Zienkiewicz and G. S. Holister (eds.) New York: Wiley, 145–197 (1965); reprinted in Int. J. Numer. Methods Eng. 52, No. 3, 287–342 (2001; Zbl 1065.74625)] and C. Johnson and the first author [Math. Comput. 38, 375–400 (1982; Zbl 0482.65058)] for which it is shown that some degrees of freedom can be eliminated without affecting the convergence rate.

In particular a model is derived which contains the average of four parameters per node and gives the quadratic convergence rate for stresses in \(L_2\). Models considered in the article are tested numerically. Numerical results, in obtaining of which the iterative version of the penalty method is used, confirm the proved rates of convergence and show that the most economical is the model of the second type.

Another class of algorithms (method II) is based on the equilibrium composite quadrilateral element considered by B. Fraeijs de Veubeke [in: Stress analysis: Recent developments in numerical and experimental methods, O. C. Zienkiewicz and G. S. Holister (eds.) New York: Wiley, 145–197 (1965); reprinted in Int. J. Numer. Methods Eng. 52, No. 3, 287–342 (2001; Zbl 1065.74625)] and C. Johnson and the first author [Math. Comput. 38, 375–400 (1982; Zbl 0482.65058)] for which it is shown that some degrees of freedom can be eliminated without affecting the convergence rate.

In particular a model is derived which contains the average of four parameters per node and gives the quadratic convergence rate for stresses in \(L_2\). Models considered in the article are tested numerically. Numerical results, in obtaining of which the iterative version of the penalty method is used, confirm the proved rates of convergence and show that the most economical is the model of the second type.

Reviewer: V. Korneev (St. Peterburg)

### 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 |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74S99 | Numerical and other methods in solid mechanics |