In this paper we propose a finite volume discretization for the threedimensional Biot poroelasticity system in multilayer domains. For stability reasons, staggered grids are used. The discretization takes into account discontinuity of the coefficients across the interfaces between layers with different physical properties. Numerical experiments based on the proposed discretization showed second order convergence in the maximum norm for the primary and flux unknowns of the system. An application example is presented as well.

In the article two finite difference schemes for the 1D poroelasticity equations (Biot model) with discontinuous coefficients are derived, analyzed, and numerically tested. A recent discretization [Gaspar et al., Appl Numer Math 44 (2003), 487-506] of these equations with constant coefficients on a staggered grid is used as a basis. Special attention is given to the interfaces and as a result a scheme with harmonic averaging of the coefficients is derived. Convergence rate of O(h 3/2 ) in a discrete H 1 -norm for both the pressure and the displacement is established in the case of an arbitrary position of the interface. Further, rate of O(h 2 ) is proven for the case when the interface coincides with a grid node. Following an approach applied to secondorder elliptic equations in [Ewing et al., SIAM J Sci Comp 23(4) (2001), 1334-1350 we derive a modified and more accurate discretization that gives second-order convergence of the fluid velocity and the stress of the solid. Finally, numerical experiments of model problems that confirm the theoretical considerations are presented.

SUMMARYGiven the continued difficulty of developing geometric multigrid methods that provide robust convergence for unstructured discretizations of compressible flow problems in aerodynamics, we turn to algebraic multigrid (AMG) as an alternative with the potential to automatically deal with arbitrary sources of stiffness on unstructured grids. We show here that AMG methods are able to solve linear problems associated with first-order discretizations of the compressible Euler equations extremely rapidly. In order to solve the linear problems resulting from second-order discretizations that are of practical interest, we employ AMG applied to the first-order system within a defect correction iteration. It is demonstrated on two-and three-dimensional test cases in a range of flow regimes (sub-, trans-and supersonic) that the described method converges rapidly and robustly.

In this paper, we present a multigrid method with problem-dependent prolongation and restriction for the three-dimensional Biot poroelasticity system in a multilayered domain. The system is discretized on a staggered grid using the finite volume method. In the discretization special care is taken of the coefficients' discontinuity. The prolongation and restriction operators are derived in a consistent manner with the discretization, so that 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 operator-dependent restriction and prolongation in the multigrid solver for the considered class of problems