Parameter-Robust Discretization and Preconditioning of Biot's Consolidation Model

@article{Lee2015ParameterRobustDA,
  title={Parameter-Robust Discretization and Preconditioning of Biot's Consolidation Model},
  author={Jeonghun J. Lee and Kent‐Andre Mardal and Ragnar Winther},
  journal={SIAM J. Sci. Comput.},
  year={2015},
  volume={39}
}
Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters ranges over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal… 

Tables from this paper

Weakly Imposed Symmetry and Robust Preconditioners for Biot’s Consolidation Model

This paper discusses the construction of robust preconditioners for finite element approximations of Biot’s consolidation model in poroelasticity based on generalizations of the Hellinger–Reissner principle of linear elasticity, where the stress tensor is one of the unknowns.

Analysis and preconditioning of parameter-robust finite element methods for Biot's consolidation model

A three-field formulation of the Biot model which has the displacement, the total pressure, and the pore pressure as unknowns is considered, and a priori estimates of the continuous problem with parameter-dependent norms are shown.

Parameter-robust stability of classical three-field formulation of Biot's consolidation model

This paper is devoted to the stability analysis of a classical three-field formulation of Biot's consolidation model where the unknown variables are the displacements, fluid flux (Darcy velocity),

New stabilized discretizations for poroelasticity and the Stokes’ equations

Robust Approximation of Generalized Biot-Brinkman Problems

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.

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

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.

New Stabilized Discretizations for Poroelasticity Equations

Two discretizations of the three-field formulation of Biot’s consolidation problem are considered: one uses Crouzeix-Raviart nonconforming linear elements; the other is based on piecewise linear elements stabilized by using face bubbles, which are subsequently eliminated.

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

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
...

References

SHOWING 1-10 OF 45 REFERENCES

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case

In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed

A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: the discrete-in-time case

In this paper, we formulate a finite element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. Here, we approximate the pressure by a mixed

Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices

Methods to handle the lack of regularity at initial times are discussed and illustrated numerically and mixed space discretization methods and a regularization method to stabilize the system and avoid locking in the pressure variable are presented.

On stability and convergence of finite element approximations of biot's consolidation problem

Stability and convergence analysis of finite element approximations of Biot's equations governing quasistatic consolidation of saturated porous media are, discussed. A family of decay functions,

A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters

This paper presents a parallel preconditioner for Biot’s equations of coupled elasticity and fluid flow in porous media based on an approximation of the exact inverse of the two-by-two block system arising from a finite element discretisation.

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

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

A Least-Squares Mixed Finite Element Method for Biot's Consolidation Problem in Porous Media

A least-squares mixed finite element method for the coupled problem of flow and deformation is presented and analyzed and Ellipticity of the corresponding variational formulation is proven for the stationary case as well as for the subproblems arising at each step of an implicit time discretization in the general time-dependent case.

A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity

In this paper, we formulate a finite-element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. We approximate the flow variables by a mixed

Analysis of Block Preconditioners for Models of Coupled Magma/Mantle Dynamics

This article formulate, analyze, and numerically test an Elman, Silvester, and Wathen-type block preconditioner for magma dynamics, and proves analytically and numericically the optimality of the preconditionser.