# 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…

## 21 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.

### 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.

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

- MathematicsJ. Comput. Phys.
- 2022

### 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.

### 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.

### Non-intrusive reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation

- Computer ScienceArXiv
- 2021

This work presents a non-intrusive model reduction framework using proper orthogonal decomposition (POD) and neural networks, based on the usual offline-online paradigm, and shows that this framework provides reasonable approximations of the DG solution, but it is significantly faster.

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