Accurate discretization of poroelasticity without Darcy stability. (English) Zbl 1470.65169
Summary: In this manuscript we focus on the question: what is the correct notion of Stokes-Biot stability? Stokes-Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes-Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes-Biot stable Euler-Galerkin discretization schemes.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74B10 | Linear elasticity with initial stresses |
| 76S05 | Flows in porous media; filtration; seepage |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35Q35 | PDEs in connection with fluid mechanics |
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