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#### Keywords

- poroelasticity (2)
- MAC type grid (1)
- finite volume discretization (1)
- interface problem (1)
- multigrid (1)
- multilayered material (1)
- operator-dependent prolongation (1)

In soil mechanics assumption of only vertical subsidence is often invoked and this leads to the one-dimensional model of poroelasticity. The classical model of linear poroelasticity is obtained by Biot [1], detailed derivation can be found e.g., in [2]. This model is applicable also to modelling certain processes in geomechanics, hydrogeology, petroleum engineering (see, e.g., [3, 8], in biomechanics (e.g., [9, 10]), in filtration (e.g., filter cake formation, see [15, 16, 17]), in paper manufacturing (e.g., [11, 12]), in printing (e.g., [13]), etc. Finite element and finite difference methods were applied by many authors for numerical solution of the Biot system of PDEs, see e.g. [3, 4, 5] and references therein. However, as it is wellknown, the standard FEM and FDM methods are subject to numerical instabilities at the first time steps. To avoid this, discretization on staggered grid was suggested in [4, 5]. A single layer deformable porous medium was considered there. This paper can be viewed as extension of [4, 5] to the case of multilayered deformable porous media. A finite volume discretization to the interface problem for the classical one-dimensional Biot model of consolidation process is applied here. Following assumptions are supposed to be valid: each of the porous layers is composed of incompressible solid matrix, it is homogeneous and isotropic. Furthermore, one of two following assumptions is valid: porous medium is not completely saturated and ﬂuid is incompressible or porous medium is completely saturated and fluid is slightly compressible. The reminder of the paper is organised as follows. Next section presents the mathematical model. Third section is devoted to the dicsretization of the continuous problem. Fourth section contains the results from the numerical experiments.

On a multigrid solver for the threedimensional Biot poroelasticity system in multilayered domains
(2006)

In this paper, we present problem–dependent prolongation and problem–dependent restriction for a multigrid solver for the three-dimensional Biot poroelasticity system, which is solved in a multilayered domain. The system is discretized on a staggered grid using the finite volume method. During the discretization, special care is taken of the discontinuous coefficients. For the efficient multigrid solver, a need in operator-dependent restriction and/or prolongation arises. We derive these operators so that they are consistent with the discretization. They account for the discontinuities of the coefficients, as well as for the coupling of the unknowns within the Biot system. A set of numerical experiments shows necessity of use of the operator-dependent restriction and prolongation in the multigrid solver for the considered class of problems.