# Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity

@article{Kumar2020ConservativeDF, title={Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity}, author={Sarvesh Kumar and Ricardo Oyarz{\'u}a and Ricardo Ruiz Baier and Ruchi Sandilya}, journal={Mathematical Modelling and Numerical Analysis}, year={2020}, volume={54}, pages={273-299} }

We introduce a numerical method for the approximation of linear poroelasticity equations, representing the interaction between the non-viscous filtration flow of a fluid and the linear mechanical response of a porous medium. In the proposed formulation, the primary variables in the system are the solid displacement, the fluid pressure, the fluid flux, and the total pressure. A discontinuous finite volume method is designed for the approximation of solid displacement using a dual mesh, whereas a…

## 20 Citations

Finite Element Solvers for Biot’s Poroelasticity Equations in Porous Media

- Computer ScienceMathematical Geosciences
- 2020

The results illustrate that three out of the five methods conserve local mass and produce similar flux approximations when conductivity alteration is included and these comparisons of the key performance indicators can be utilized to choose the preferred method based on the required accuracy and the available computational resources.

Parameter-robust methods for the Biot-Stokes interfacial coupling without Lagrange multipliers

- MathematicsSSRN Electronic Journal
- 2021

This paper proposes robust preconditioners for this perturbed saddle-point problem using appropriately weighted operators in fractional Sobolev and metric spaces at the interface and proposes a five-field mixed-primal finite element scheme solving for Stokes velocitypressure and Biot displacement-total pressure-fluid pressure.

Virtual element methods for the three-field formulation of time-dependent linear poroelasticity

- EngineeringAdv. Comput. Math.
- 2021

Under standard assumptions on the computational domain, optimal a priori error estimates are established and these estimates are valid independently of the values assumed by the dilation modulus and the specific storage coefficient, implying that the formulation is locking-free.

A four-field mixed finite element method for Biot's consolidation problems

- MathematicsElectronic Research Archive
- 2021

This article presents a four-field mixed finite element method for Biot's consolidation problems, where the four fields include the displacement, total stress, flux and pressure for the porous medium…

Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model

- MathematicsESAIM: Mathematical Modelling and Numerical Analysis
- 2021

We present an a priori and a posteriori error analysis of a conforming finite element method for a four-field formulation of the steady-state Biot’s consolidation model. For the a priori error…

The Biot-Stokes coupling using total pressure: formulation, analysis and application to interfacial flow in the eye

- EngineeringComputer Methods in Applied Mechanics and Engineering
- 2022

Twofold saddle-point formulation of Biot poroelasticity with stress-dependent diffusion

- MathematicsArXiv
- 2021

A stress/total-pressure formulation for poroelasticity that includes the coupling with steady nonlinear diffusion modified by stress can be used to study waste removal in the brain parenchyma, where diffusion of a tracer alone or a combination of advection and diffusion are not sufficient to explain the alterations in rates of filtration observed in porous media samples.

Well-posedness and discrete analysis for advection-diffusion-reaction in poroelastic media

- MathematicsApplicable Analysis
- 2021

A PDE system modelling poromechanical processes interacting with diffusing and reacting solutes in the medium is analysed and the well-posedness of the nonlinear set of equations is investigated using fixed-point theory, Fredholm's alternative, a priori estimates, and compactness arguments.

Robust approximation of generalized Biot-Brinkman problems

- EngineeringArXiv
- 2021

This paper introduces, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized BiotBrinkman equations and demonstrates that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes.

Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot's equations utilizing total pressure

- MathematicsSIAM J. Sci. Comput.
- 2021

The stability of the continuous variational mixed problems, which hinges upon using adequately weighted spaces, is addressed in detail; and the efficacy of the proposed preconditioners, as well as their robustness with respect to relevant material properties, is demonstrated through several numerical experiments.

## References

SHOWING 1-10 OF 64 REFERENCES

A stabilized finite element method for finite-strain three-field poroelasticity

- MathematicsComputational mechanics
- 2017

A stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium that is stable in both the limiting cases of small and large permeability and enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions.

Stabilized Lowest-Order Finite Element Approximation for Linear Three-Field Poroelasticity

- MathematicsSIAM J. Sci. Comput.
- 2015

A stabilized conforming mixed finite element method for the three-field poroelasticity problem is developed and analyzed, using the lowest possible approximation order, namely piecewise constant approximation for the pressure and piecewise linear continuous elements for the displacements and fluid flux.

A locally conservative stabilized continuous Galerkin finite element method for two-phase flow in poroelastic subsurfaces

- EngineeringJ. Comput. Phys.
- 2017

A Mixed Finite Element Method for Nearly Incompressible Multiple-Network Poroelasticity

- Computer ScienceSIAM J. Sci. Comput.
- 2019

This paper presents and analyzes a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity equations and shows that this formulation is robust in the limits of incompressibility, vanishing storage coefficients, and vanishing transfer between networks.

A coupling of weak Galerkin and mixed finite element methods for poroelasticity

- MathematicsComput. Math. Appl.
- 2017

A Lagrange multiplier method for a Stokes–Biot fluid–poroelastic structure interaction model

- EngineeringNumerische Mathematik
- 2018

A finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium and the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters is studied.

A nonconforming finite element method for the Biot's consolidation model in poroelasticity

- GeologyJ. Comput. Appl. Math.
- 2017

Analysis of a multiphysics finite element method for a poroelasticity model

- Engineering
- 2018

This article concerns finite element approximations of a quasi-static poroelasticity model in displacement– pressure formulation, which describes the dynamics of poro-elastic materials under an…

A coupling of nonconforming and mixed finite element methods for Biot's consolidation model

- Mathematics
- 2013

In this article, we develop a nonconforming mixed finite element method to solve Biot's consolidation model. In particular, this work has been motivated to overcome nonphysical oscillations in the…

Mixed finite element - discontinuous finite volume element discretization of a general class of multicontinuum models

- Computer ScienceJ. Comput. Phys.
- 2016