Stability and Convergence of Two Three-Field Finite Element Formulations for Elasticity

Abstract.

In this work analyses are carried out to verify the well-posedness of two three-field formulations for linear elasticity that have been shown to work well in practice, but for which analyses are absent. Both are formulated in three dimensions on tetrahedral elements. The first has as primary unknowns the displacement, pressure, and dilatation, and the second is a formulation with primary unknowns the displacement, pressure and enhanced strain, with the space of enhanced strains spanned by functions that are bubble functions on a surface of the element. It is shown that the first-mentioned formulation is a special case of a general three-field formulation, for which well-posedness conditions have been established. These conditions are used here to show that the pressure-dilatation-displacement formulation is uniformly convergent in the incompressible limit. The pressure-enhanced strain-displacement formulation is not amenable to such an approach; instead, it is reformulated in the form of a standard discrete saddle point problem, and the uniform convergence of the formulation established directly by verifying the ellipticity and inf-sup conditions.