Abstract

The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns. The first is the classical pressure instability that occurs when the solid is incompressible, whereas the second is the lack of stability in the stresses. To overcome these instabilities, there are two options. The first is to use different interpolation for all the unknowns satisfying two inf-sup conditions. Whereas there are several displacement-pressure interpolations that render the pressure stable, less possibilities are known for the stress interpolation. The second option is to use a stabilized finite element formulation instead of the plain Galerkin approach. If this formulation is properly designed, it is possible to use arbitrary interpolation for all the unknowns. The purpose of this paper is precisely to present one of such formulations. In particular, it is based on the decomposition of the unknowns into their finite element component and a subscale, which will be approximated and whose goal is to yield a stable formulation. A singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space. We describe the design of the formulation and present the results of its numerical analysis.
 

Abstract

In this work we propose a stabilized nite element method that permits us to circumvent discrete inf-sup conditions, e.g. allowing equal order interpolation. The type of method we propose belongs to the family of symmetric stabilization techniques, which are based on the introduction of additional terms that penalize the di erence between some quantities, i.e. the pressure gradient in the Stokes problem, and their nite element projections. The key feature of the formulation we propose is the de nition of the projection to be used, a non-standard Scott-Zhang projector that is well-de ned for L1() functions. The resulting method has some appealing features: the projector is local and nested meshes or enriched spaces are not required.