A family of higher order mixed finite element methods for plane elasticity. (English) Zbl 0558.73066

The authors construct a new family of finite elements for the approximation of a mixed variational formulation of linear elasticity (formulation in terms of displacements and stresses). Approximation properties of these elements are studied and estimates of optimal order are derived for both, the displacement and stress field. All estimates are valid for the incompressible case, as well. Elements are the vector analogue of Raviart-Thomas mixed finite elements for the scalar elliptic equation.
Reviewer: J.Haslinger


74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S99 Numerical and other methods in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
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[1] Babu?ka, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. In: The mathematical foundations of the finite element method with applications to partial differential equations (A.K. Aziz, ed.). New York: Academic Press 1972 · Zbl 0268.65052
[2] Brezzi, F.: On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. R.A.I.R.O. Anal. Num?r.2, 129-151 (1974) · Zbl 0338.90047
[3] Douglas, J., Dupont, T., Percell, P., Scott, R.: A family ofC 1 finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. R.A.I.R.O. Anal. Num?r.13, 227-255 (1979) · Zbl 0419.65068
[4] Douglas, J., Roberts, J.E.: Mixed finite elements methods for second order elliptic problem. Matem?tica Applicade e Computacional1, 91-103 (1982) · Zbl 0482.65057
[5] Douglas, J., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. (To appear in Math. Comput.) · Zbl 0624.65109
[6] Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces Math. Comput.34, 441-463 (1980) · Zbl 0423.65009
[7] Falk, R.S., Osborn, J.E.: Error estimates for mixed methods. R.A.I.R.O., Anal. Num?r.14, 309-324 (1980) · Zbl 0467.65062
[8] Johnson, C., Mercier, B.: Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math.30, 103-116 (1978) · Zbl 0427.73072
[9] Percell, P.: On cubic and quartic Clough-Tocher finite elements. SIAM J. Numer. Anal.13, 100-103 (1976) · Zbl 0319.65064
[10] Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. Mathematical aspects of the finite element method. Lecture Notes in Mathematics 606. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0362.65089
[11] Scholz, R.:L ?-convergence of saddle-point approximation for second order problems. R.A.I.R.O. Anal. Num?r.11, 209-216 (1977) · Zbl 0356.35026
[12] Scholz, R.: A remark on the rate of convergence for a mixed finite element method for second order problems. Numer. Funct. Anal. and Optimiz.4, 269-277 (1981/82) · Zbl 0481.65066
[13] Scholz, R.: OptimalL ?-estimates for a mixed finite element method for second order elliptic and parabolic problems. (To appear in Calcolo) · Zbl 0571.65092
[14] Temam, R.: Navier-stokes equations. Amsterdam: North Holland 1977 · Zbl 0383.35057
[15] Thomas, J.M.: Sur l’analyse num?rique des m?thodes d’?l?ments finis mixtes et hybrides. Th?se, Paris 1977
[16] Vogelius, M.: An analysis of thep-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal order error estimates. Numer. Math.41, 39-53 (1983) · Zbl 0504.65061
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