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On convergence of certain finite volume difference discretizations for 1D poroelasticity interface problems

: Ewing, R.E.; Iliev, O.P.; Lazarov, R.D.; Naumovich, A.


Numerical methods for partial differential equations 23 (2007), No.3, pp.652-671
ISSN: 0749-159X
Journal Article
Fraunhofer ITWM ()

In the article two finite difference schemes for the ID 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 0 (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.