A stability condition for the numerical simulation of poroelastic systems

In the numerical simulation of the Biot model numerical oscillations may occur if the spatial and discretization parameters are not properly chosen. When the time-step is too small, the grid has to be fine enough to resolve the boundary layers that appear in the early stages of the simulation. In this work, we extend a strategy that has been successfully employed in the one-dimensional case for the detection of a critical time-step, below which instabilities appear. The idea is to study under which condition the Schur complement of the poroelastic system enjoys a discrete maximum principle property. Differently from the one-dimensional case, in the vectorial case the shear modulus plays a fundamental role for the computation of the Schur complement and the maximum principle argument can be applied only for small shear. We will also investigate the sign of the entries of the Schur complement when the shear and the bulk modulus are of the same magnitude.