A Nonconforming Finite Element Method for the Biot's Consolidation Model in Poroelasticity

TL;DR

This paper introduces a stable, low-order finite element scheme for Biot's poroelasticity model that avoids pressure oscillations and reduces computational costs through mass lumping, with proven convergence and numerical validation.

Contribution

It develops a novel finite element method combining Crouzeix-Raviart and Raviart-Thomas-Nedelec elements with mass lumping for efficient, stable simulation of Biot's model.

Findings

The scheme converges reliably in numerical tests.

Mass lumping effectively prevents pressure oscillations.

The method reduces computational costs while maintaining accuracy.

Abstract

A stable finite element scheme that avoids pressure oscillations for a three-field Biot's model in poroelasticity is considered. 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-Nedelec elements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass lumping technique is introduced for the Raviart-Thomas-Nedelec elements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost. We show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass lumping. Finally, numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the Raviart-Thomas-Nedelec elements is used.