New stable mixed finite element approximations for problems in linear elasticity

We construct stable and convergent finite element approximations for two- and three-dimensional problems in elasticity. The approximations are based on three-field mixed variational formulations, and are uniformly convergent in the incompressible limit. The basis for the constructions is a set of general sufficient conditions established in Lamichhane, Reddy and Wohlmuth, Numer. Math. 104 (2006). In that work the focus was on low-order quadrilateral elements. Here it is shown that the methodology is readily extended to families of elements that are built on stable velocity–pressure pairs for the Stokes problem. Various numerical examples serve to illustrate the convergence behaviour of the new elements.

► We construct new elements for displacement–stress–strain formulations in elasticity.
► The new elements are derived from stable Stokes velocity-pressure pairs.
► The new formulations are shown to be convergent in the incompressible limit.
► A number of illustrative numerical examples are presented.