When it comes to the analysis of fully saturated porous materials, a plethora of formulations  are available in the literature. Among them, the so-called u − p formulation constitutes a well-established approach when low-frequency problems are considered. It can, however, exhibit instabilities due to the discretisation as primary variables of physically different quantities, i.e., a displacement and a pressure field. Two techniques for dealing with this instability (the Taylor-Hood family of elements and the Polynomial Pressure Projection) are summarised and cross-compared for the Material Point Method and the Generalised Interpolation Material Point Method.
Pretti, G., Coombs, W., & Augarde, C. (in press). A comparison of approaches for modelling poro-mechanics in the Material Point Method.