European Union’s Horizon 2020 research and innovation programme

Marie Sklodowska-Curie grant agreement NO 705402, POROSOS

STABLE DISCRETIZATIONS FOR POROELASTICITY

We consider the popular P1-RT0-P0 discretization of the three-field formulation of Biot’s consolidation problem. It is well-known that this finite-element formulation does not satisfy an inf-sup condition uniformly with respect to the physical parameters. 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.

PUBLICATIONS:

New stabilized discretizations for poroelasticity and the Stokes’ equations. C. Rodrigo, X. Hu, P. Ohm, J.H. Adler, F.J. Gaspar, L.T. Zikatanov. Computer Methods in Applied Mechanics and Engineering 341. 467-484 (2018) : LINK

Previous related works:

Stability and monotonicity for some discretizations of the Biot’s consolidation model. C. Rodrigo, F.J. Gaspar, X. Hu, L. Zikatanov. Computer Methods in Applied Mechanics and Engineering 298, 183-204 (2016): LINK

We consider finite element discretizations of the Biot’s consolidation model 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.

A noncoforming finite element method for the Biot’s consolidation model in poroelasticity. X. Hu, C. Rodrigo, F.J. Gaspar, L. Zikatanov. Journal of Computational and Applied Mathematics (2017).: LINK

We consider a stable finite element scheme that avoids pressure oscillations for a three-field Biot’s model. The involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: Crouzeix–Raviart finite elements for the displacements, lowest order Raviart–Thomas-Nédélec elements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass-lumping technique is introduced for the Raviart–Thomas-Nédélec elements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost.