Stability and Monotonicity for Some Discretizations of the Biot's Model

TL;DR

This paper analyzes finite element discretizations of Biot's model, addressing stability and monotonicity issues, especially pressure oscillations, and introduces stabilization techniques validated by numerical results.

Contribution

It introduces a stabilization method for finite element discretizations of Biot's model that ensures pressure monotonicity and removes non-physical oscillations.

Findings

Stability and convergence of discretizations are established.

Stabilization effectively removes pressure oscillations.

Numerical results confirm improved monotonicity.

Abstract

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.