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.

A. Ouazzi, S. Turek, H. Damanik. A curvature-free multiphase flow solver via surface stress-based formulation. Int J Numer Meth Fluids 88(1) (2018) DOI 10.1002/fld.4509

E. Hachem, S. Feghali, R. Codina, T. Coupez. Immersed stress method for fluid-structure interaction using anisotropic mesh adaptation. Int. J. Numer. Meth. Engng 94(9) (2013) DOI 10.1002/nme.4481

R. Codina, J. Baiges. Weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non-matching meshes. Int. J. Numer. Meth. Engng 104(7) (2014) DOI 10.1002/nme.4815

E. Hachem, S. Feghali, T. Coupez, R. Codina. A three-field stabilized finite element method for fluid-structure interaction: elastic solid and rigid body limit. Int. J. Numer. Meth. Engng 104(7) (2015) DOI 10.1002/nme.4972

S. Badia, R. Codina. Unified Stabilized Finite Element Formulations for the Stokes and the Darcy Problems. SIAM J. Numer. Anal. 47(3) DOI 10.1137/08072632x

G. Barrenechea, E. Castillo, R. Codina. Time-dependent semidiscrete analysis of the viscoelastic fluid flow problem using a variational multiscale stabilized formulation. 39(2) (2018) DOI 10.1093/imanum/dry018

C. Bayona Roa, J. Baiges, R. Codina. Variational multi-scale finite element approximation of the compressible Navier-Stokes equations. Int Jnl of Num Meth for HFF 26(3/4) DOI 10.1108/hff-11-2015-0483

R. Codina. On the Design of Algebraic Fractional Step Methods for Viscoelastic Incompressible Flows. (2018) DOI 10.1007/978-3-319-97613-6_6