×

zbMATH — the first resource for mathematics

On the construction of optimal mixed finite element methods for the linear elasticity problem. (English) Zbl 0563.65072
We consider the mixed formulation of the finite element method for linear elasticity, i.e. a formulation where independent approximations are used for both the stresses and the displacements. In the first part of the paper the application of the Babuška-Brezzi theory to this problem is discussed and it is shown, that the use of certain mesh-dependent norms simplifies the analysis considerably. The remaining of the paper is devoted to the construction of mixed methods with optimal convergence rates. A systematic way of constructing such methods is proposed and finally the ideas are applied in some examples.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
74B10 Linear elasticity with initial stresses
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Arnold, D.N., Douglas, J., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math.45, 1-22 (1984) · Zbl 0558.73066
[2] Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Preprint, University of Pavia, 1983 · Zbl 0593.76039
[3] Babu?ka, I.: The finite element method with Lagrangian multipliers. Numer. Math.20, 179-192 (1973) · Zbl 0258.65108
[4] Babu?ka, I., Aziz, A.: Survey lectures on the mathematical foundations of the finite element method. In: The Mathematical Foundations of the Finite Element Method with application to Partial Differential Equations (A.K. Aziz ed), pp. 5-359. New York: Academic Press 1973
[5] Babu?ka, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput.35, 1039-1062 (1980) · Zbl 0472.65083
[6] Bercovier, M.: Perturbation of mixed variational problems. Application to mixed finite element methods. RAIRO, Anal. Numér.12, 211-236 (1978) · Zbl 0428.65059
[7] Boland, J.M., Nicolaides, R.A.: Stability of finite elements under divergence constraints. SIAM J. Numer. Anal.20, 722-731 (1983) · Zbl 0521.76027
[8] Brezzi, F.: On the existence uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Ser. Rouge8, 129-151 (1974) · Zbl 0338.90047
[9] Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. REPORT-MAT-A 219, Helsinki University of Technology, 1984
[10] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[11] Clemént, P.: Approximation by finite elements using local regularization. RAIRO Ser. Rouge9, 77-84 (1975)
[12] Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Ser. Rouge7, 33-76 (1973) · Zbl 0302.65087
[13] Johnson, C., Mercier, B.: Some equilibrium finite element methods for two-dimensional elasticity problems. Numer. Math.30, 103-116 (1978) · Zbl 0427.73072
[14] Karp, S.N., Karal, F.C.: The elastic-field in the neighbourhood of a crack of arbitrary angle. CPAM15, 413-421 (1962) · Zbl 0166.20705
[15] Mansfield, L.: Finite element subspaces with optimal rate of convergence for the stationary Stokes problem. RAIRO, Anal. Numér.16, 49-66 (1982) · Zbl 0477.65084
[16] Mirza, F.A., Olson, M.D.: The mixed finite element method in plane elasticity. Int. J. Numer. Methods Eng.15, 273-290 (1980) · Zbl 0426.73069
[17] Pitkäranta, J., Stenberg, R.: Analysis of some mixed finite element methods for plane elasticity equations. Math. Comput.41, 399-423 (1983) · Zbl 0537.73057
[18] Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Proceedings of the Symposium on the Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics 606, pp. 292-315. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0362.65089
[19] Stenberg, R.: Analysis of mixed finite element methods for the Stokes problem: A unified approach. Math. Comput.42, 9-23 (1984) · Zbl 0535.76037
[20] Williams, M.L.: Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. Appl. Mech.19, 526-528 (1952)
[21] Zienkiewicz, O.C.: The Finite Element Method, 3rd edition. New York: McGraw-Hill 1977 · Zbl 0435.73072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.