Stability and Monotonicity for Some Discretizations of the Biot's Model
For researchers in poroelasticity and numerical methods, this work addresses the practical issue of non-physical pressure oscillations, offering a stabilization technique, though the analysis is limited to 1D and specific element types.
The paper analyzes finite element discretizations of Biot's consolidation model, showing that Stokes-stable elements fail to provide monotone pressure approximations in low-permeability or small-time-step regimes, and introduces a stabilization term that removes oscillations, with numerical results confirming monotonicity.
We consider finite element discretizations of the Biot's consolidation model in poroelasticity with MINI and stabilized P1-P1 elements. We analyze the convergence of the fully discrete model based on spatial discretization with these types of finite elements and implicit Euler method in time. We also address the issue related to the presence of non-physical oscillations in the pressure approximation for low permeabilities and/or small time steps. We show that even in 1D a Stokes-stable finite element pair fails to provide a monotone discretization for the pressure in such regimes. We then introduce a stabilization term which removes the oscillations. We present numerical results confirming the monotone behavior of the stabilized schemes.