Upper bounds of the error in local quantities using equilibrated and compatible finite element solutions for linear elastic problems. (English) Zbl 1086.74041

Summary: When local quantities are computed using the principle of virtual work, dual analysis, which provides an upper bound of the global error, may also be applied to the virtual problem. H. Greenberg [J. Math. Phys. 27, 161–182 (1948; Zbl 0031.31001)] and K. Washizu [J. Math. Phys. 32, 117–128 (1953; Zbl 0051.41004)] proposed alternative approaches to combine the global error bounds of the real and virtual problems, providing upper bounds of the local error. It is shown in this paper that optimising Greenberg’s approach corresponds to using Washizu’s approach, which, in turn, may be further improved. These approaches are used to provide finite element error indicators for adaptive refinement.


74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
65N15 Error bounds for boundary value problems involving PDEs
Full Text: DOI


[1] Greenberg, H., The determination of upper and lower bounds for the solution of the Dirichlet problem, J. math. phys., 27, 161-182, (1948) · Zbl 0031.31001
[2] Washizu, K., Bounds for solutions of boundary value problems in elasticity, J. math. phys., 32, 117-128, (1953) · Zbl 0051.41004
[3] Fraeijs de Veubeke, B., Displacement and equilibrium models in the finite element method, () · Zbl 0359.76021
[4] E. Trefftz, Ein Gegenstück zum Ritzschen Verfahren, in: Proceedings of the 2nd International Congress of Applied Mechanics, 1926, pp. 131-137.
[5] Prager, W.; Synge, J., Approximations in elasticity based on the concept of function space, Quart. appl. math., 5, 3, 241-269, (1947) · Zbl 0029.23505
[6] J. Debongnie, A general theory of dual error bounds by finite elements, Tech. Rep. LMF/D5, University of Liège, 1983.
[7] Ladeveze, P.; Leguillon, D., Error estimate procedure in the finite element method and applications, SIAM J. numer. anal., 20, 3, 483-509, (1983) · Zbl 0582.65078
[8] Pereira, O.; Almeida, J.; Maunder, E., Adaptive methods for hybrid equilibrium finite element models, Comput. methods appl. mech. engrg., 176, 19-39, (1999) · Zbl 0991.74072
[9] Turner, M.; Clough, R.; Martin, H.; Topp, L., Stiffness and deflection analysis of complex structures, J. aeronaut. sci., 23, 9, 805-854, (1956) · Zbl 0072.41003
[10] Yang, D.; Kelly, W.; Isles, J., A posteriori pointwise upper bound estimates in the finite element error, Int. J. numer. methods engrg., 36, 1279-1298, (1993) · Zbl 0772.73085
[11] Albanese, R.; Fresa, R., Upper and lower bounds for local electromagnetic quantities, Int. J. numer. methods engrg., 42, 499-515, (1998) · Zbl 0915.65124
[12] Babuska, I.; Miller, A., The post-processing approach in the finite element method (parts 1, 2 and 3), Int. J. numer. methods engrg., 20, 1085-1129, (1984), 2311-2324
[13] Babuska, I.; Strouboulis, T.; Upadhyay, C.; Gangaraj, S., A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method, Int. J. numer. methods engrg., 38, 4207-4235, (1995) · Zbl 0844.65078
[14] Babuska, I.; Strouboulis, T.; Gangaraj, S., Guaranteed computable bounds for the exact error in the finite element solution. part I: one-dimensional model problem, Comput. methods appl. mech. engrg., 176, 1-4, 51-79, (1999) · Zbl 0936.65094
[15] Strouboulis, T.; Babuska, I.; Gangaraj, S.; Copps, K.; Datta, D., A posteriori estimation of the error in the error estimate, Comput. methods appl. mech. engrg., 176, 1-4, 387-418, (1999) · Zbl 0948.74068
[16] Strouboulis, T.; Babuska, I.; Gangaraj, S., Guaranteed computable bounds for the exact error in the finite element solution—part II: bounds for the energy norm of the error in two dimensions, Int. J. numer. methods engrg., 47, 1-3, 427-475, (2000) · Zbl 0962.65069
[17] Rannacher, R.; Suttmeier, F., A feed-back approach to error control in finite element methods: application to linear elasticity, Comput. mech., 19, 434-446, (1997) · Zbl 0894.73170
[18] Peraire, J.; Patera, A., Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement, (), 199-216
[19] Ladevèze, P.; Rougeot, P.; Blanchard, P.; Moreau, J., Local error estimators for finite element linear analysis, Comput. methods appl. mech. engrg., 176, 1-4, 231-246, (1999) · Zbl 0967.74066
[20] Prudhomme, S.; Oden, J.T.; Westermann, T.; Bass, J.; Botkin, M., Practical methods for a posteriori error estimation in engineering applications, Int. J. numer. methods engrg., 56, 8, 1193-1224, (2003) · Zbl 1038.74045
[21] Pereira, O.; Almeida, J., Majorantes do erro EM grandezas locais obtidas a partir de soluções duais de elementos finitos, ()
[22] Pereira, O.; Almeida, J., Adaptive refinement for a local error bound based on duality, Comput. assisted mech. engrg. sci., 10, 4, 565-574, (2003) · Zbl 1136.74381
[23] Almeida, J.; Pereira, O., Equilibrium solutions for error estimation and adaptivity. why, how and at what cost?, ()
[24] Jones, R., A generalization of the direct-stiffness method of structural analysis, Aiaa j., 2, 5, 821-826, (1964) · Zbl 0119.19304
[25] Pian, T.; Tong, P., Basis of finite element methods for solid continua, Int. J. numer. methods engrg., 1, 3-28, (1969) · Zbl 0252.73052
[26] Almeida, J.; Freitas, J., Continuity conditions for finite element analysis of solids, Int. J. numer. methods engrg., 33, 845-853, (1992) · Zbl 0825.73828
[27] Almeida, J.; Freitas, J., Alternative approach to the formulation of hybrid equilibrium finite elements, Comput. struct., 40, 4, 1043-1047, (1991)
[28] Washizu, K., Variational methods in elasticity and plasticity, (1975), Pergamon Press Oxford · Zbl 0164.26001
[29] Babuska, I.; Rheinboldt, W., A posteriori error estimates for the finite element method, Int. J. numer. methods engrg., 12, 1597-1615, (1978) · Zbl 0396.65068
[30] Kelly, D.; Gago, J.; Zienkiewicz, O., A posteriori error analysis and adaptive processes in the finite element method: part I—error analysis, Int. J. numer. methods engrg., 19, 1593-1619, (1983) · Zbl 0534.65068
[31] Zienkiewicz, O.; Zhu, J., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. numer. methods engrg., 24, 337-357, (1987) · Zbl 0602.73063
[32] Zienkiewicz, O.; Zhu, J., The superconvergent patch recovery (SPR) and adaptive finite element refinement, Comput. methods appl. mech. engrg., 101, 207-224, (1992) · Zbl 0779.73078
[33] Pereira, O.; Bugeda, G., Mesh optimality criteria and remeshing strategies for singular point problems, ()
[34] Jakobsen, B., The sleipner accident and its causes, (), 102-108
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