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… 
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20 Citations
Background Citations
6
Methods Citations
8

20 Citations

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Virtual element methods for the three-field formulation of time-dependent linear poroelasticity
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A four-field mixed finite element method for Biot's consolidation problems
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Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model
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
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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
TLDR
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
TLDR
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
TLDR
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.
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