New stabilized discretizations for poroelasticity and the Stokes’ equations

Abstract

In this work, we consider the popular P1-RT0-P0 discretization of the three-field formulation of Biot's consolidation problem. Since this finite-element formulation does not satisfy an inf-sup condition uniformly with respect to the physical parameters, several issues arise in numerical simulations. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur. Thus, we propose a stabilization technique that enriches the piecewise linear finite-element space of the displacement with the span of edge/face bubble functions. We show that for Biot's model this does give rise to discretizations that are uniformly stable with respect to the physical parameters. We also propose a perturbation of the bilinear form, which allows for local elimination of the bubble functions and provides a uniformly stable scheme with the same number of degrees of freedom as the classical P1-RT0-P0 approach. We prove optimal stability and error estimates for this discretization. Finally, we show that this scheme can also be successfully applied to Stokes' equations, yielding a discrete problem with optimal approximation properties and with minimum number of degrees of freedom (equivalent to a P1-P0 discretization). Numerical tests confirm the theory for both poroelastic and Stokes' test problems.

On the optimization of the fixed‐stress splitting for Biot's equations

“…18 For a discussion on the stability of different spatial discretizations, we refer to the recent papers. 19,20 Independently of the chosen discretization, there are two popular alternatives for solving Biot's equations: monolithically or by using an iterative splitting algorithm. The former has the advantage of being unconditionally stable, whereas a splitting method is much easier to implement, typically building on already available, tailored, separate numerical codes for porous media flow and for mechanics.…”

Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models

“…There are various discretizations for the classic three‐field formulation of Biot's model that fit in the framework of full parameter‐robust stability analysis presented in the work of Hong et al For example, the triplets together with the stabilization techniques suggested in the works of Hansbo et al and Hu et al (see also the work of Fortin et al); the triplets (in 2D) and (in 3D); ; the stabilized discretization, recently advocated in the work of Rodrigo et al; or the finite element methods proposed in the work of Lee would qualify for such parameter robustness. Coupling continuous or discontinuous Galerkin (DG) approximations of the solid displacement with a mixed method for the pressure, error estimates were obtained in the works of Phillips et al Following the theoretical framework presented in this paper, these discretizations can be applied to the MPET model.…”

A coupling of hybrid mixed and continuous Galerkin finite element methods for poroelasticity

“…However, the triple P1-RT0-P0 does not satisfy Biot-Stokes stability condition uniformly with respect to the discretization and physical parameters of the problem [2,31]. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur [31].…”

“…However, the triple P1-RT0-P0 does not satisfy Biot-Stokes stability condition uniformly with respect to the discretization and physical parameters of the problem [2,31]. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur [31]. Due to the same reason, the hybridized P1-RT0-P0 scheme presented in [1] has the same stability issue, i.e., locking phenomena may occur when the permeability is small with respect to the mesh size.…”

“…Due to the same reason, the hybridized P1-RT0-P0 scheme presented in [1] has the same stability issue, i.e., locking phenomena may occur when the permeability is small with respect to the mesh size. In [31], a stabilization technique is introduced to P1-RT0-P0 based on bubble enrichment to overcome the stability issue. Here, we apply the same stabilization technique, enriching the piecewise linear finite element space with edge/face bubble functions for displacement, to stabilize the hybrid P1-RT0-P0 scheme developed in [1].…”