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On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems. (English) Zbl 1142.74386

Summary: Two finite difference schemes for the 1D poroelasticity equations (Biot model) with discontinuous coefficients are derived, analyzed, and numerically tested. A recent discretization [F. J. Gaspar et al., Appl. Numer. Math. 44, No. 4, 487–506 (2003; Zbl 1023.76032)] 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 second-order elliptic equations in [Ewing et al., SIAM J. Sci. Comput. 23, No. 4, 1334-1350 (2001)] 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.

MSC:

74S10 Finite volume methods applied to problems in solid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76S05 Flows in porous media; filtration; seepage

Citations:

Zbl 1023.76032
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References:

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