On Stabilized Finite Element Methods for Linear Systems of Convection-Diffusion-Reaction Equations

The accurate prediction of the behavior of geostructures is based on the strong coupling between the pore fluid and the solid skeleton. If the relative acceleration of the fluid phase to the skeleton is neglected, the equations describing the problem can be written in terms of skeleton displacements (or velocities) and pore pressures.
This mixed problem is similar to others found in solid and fluid dynamics. In the limit case of zero permeability and incompressibility of the fluid phase, the restrictions on the shape functions used to approximate displacements and pressures imposed by Babuska-Brezzi conditions or the Zienkiewicz-Taylor patch test hold.
As a consequence, it is not possible to use directly elements with the same order of interpolation for the field variables .
This paper proposes two alternative methods allowing us to circunvent the BB restrictions in the incompressibility limit, making it possible to use elements with the same order of interpolation. The first consists on introducing the divergence of the momentum equation of the mixture as an stabilization term, the second is a generalization of the two step-projection method introduced by Chorin for fluid dynamics problems.