Differential complexes and stability of finite element methods. II: The elasticity complex. (English) Zbl 1119.65399
Arnold, Douglas N. (ed.) et al., Compatible spatial discretizations. Papers presented at IMA hot topics workshop: compatible spatial discretizations for partial differential equations, Minneapolis, MN, USA, May 11–15, 2004. New York, NY: Springer (ISBN 0-387-30916-0/hbk). The IMA Volumes in Mathematics and its Applications 142, 47-67 (2006).
Summary: [For part I see ibid. 142, 23–46 (2006; Zbl 1119.65398).]
A close connection between the ordinary de Rham complex and a corresponding elasticity complex is utilized to derive new mixed finite element methods for linear elasticity. For a formulation with weakly imposed symmetry, this approach leads to methods which are simpler than those previously obtained. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field. We also discuss how the strongly symmetric methods proposed by D. N. Arnold and R. Winther [Numer. Math. 92, No. 3, 401–419 (2002; Zbl 1090.74051)] can be derived in the present framework. The method of construction works in both two and three space dimensions, but for simplicity the discussion here is limited to the two dimensional case.
For the entire collection see [Zbl 1097.65003].
A close connection between the ordinary de Rham complex and a corresponding elasticity complex is utilized to derive new mixed finite element methods for linear elasticity. For a formulation with weakly imposed symmetry, this approach leads to methods which are simpler than those previously obtained. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field. We also discuss how the strongly symmetric methods proposed by D. N. Arnold and R. Winther [Numer. Math. 92, No. 3, 401–419 (2002; Zbl 1090.74051)] can be derived in the present framework. The method of construction works in both two and three space dimensions, but for simplicity the discussion here is limited to the two dimensional case.
For the entire collection see [Zbl 1097.65003].
MSC:
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74B05 | Classical linear elasticity |
