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

Abstract

AbstractIn 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(h3/2) in a discrete H1‐norm for both the pressure and the displacement is established in the case of an arbitrary position of the interface. Further, rate of O(h2) 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 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. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007