  The Material Point Method (MPM) is widely used to analyse coupled
(solid-water) problems under large deformations/displacements. However, if not
addressed carefully, MPM u-p formulations for poro-mechanics can be affected by
two major sources of instability. Firstly, inf-sup condition violation can
arise when the spaces for the displacement and pressure fields are not chosen
correctly, resulting in an unstable pressure field. Secondly, the intrinsic
nature of particle-based discretisation makes the MPM an unfitted mesh-based
method, which can affect the system's condition number and solvability,
particularly when background mesh elements are poorly populated. This work
proposes a solution to both problems. The inf-sup condition is avoided using
two overlapping meshes, a coarser one for the pressure and a finer one for the
displacement. This approach does not require stabilisation of the primary
equations since it is stable by design and is particularly valuable for
low-order shape functions. As for the system's poor condition number, a face
ghost penalisation method is added to both the primary equations, which
constitutes a novelty in the context of MPM mixed formulations. This study
frequently makes use of the theories of functional analysis or the unfitted
Finite Element Method (FEM). Although these theories may not directly apply to
the MPM, they provide a robust and logical basis for the research. These
rationales are further supported by three numerical examples, which encompass
both elastic and elasto-plastic cases and drained and undrained conditions.
