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A mixed finite element method for elasticity in three dimensions. (English) Zbl 1125.74382
From the text: We describe a stable mixed finite element method for linear elasticity in three dimensions. This answers an assertion of D. N. Arnold [Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. I: Plenary lectures and ceremonies. Beijing: Higher Education Press. 137–157 (2002; Zbl 1023.65113)]: ‘Four decades of searching for mixed finite elements for elasticity, beginning in the 1960s, did not yield any stable elements with polynomial shape functions’. First D. N. Arnold and R. Winther [Numer. Math. 92, No. 3, 401–419 (2002; Zbl 1090.74051)] gave stable elements for a two-dimensional version of the problem, and remarked that the method of construction works in both two and three space dimensions, but for simplicity the discussion was limited to the two-dimensional case.

MSC:
74S05Finite element methods in solid mechanics
74B05Classical linear elasticity
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
References:
[1]Arnold, D. (2002). Differential complexes and numerical stability. In: Proceedings of the International Congress of Mathematicians, Vol.I (Beijing, 2002), 137–157, Higher Ed. Press, Beijing.
[2] · Zbl 1090.74051 · doi:10.1007/s002110100348
[3]Arnold, D., Awanou, G. and Winther, R. A family of mixed elements for elasticity in three dimensions. In preparation.