Accurate discretization of poroelasticity without Darcy stability Stokes–Biot stability revisited

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 converge...

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Bibliographic Details
Published inBIT Numerical Mathematics Vol. 61; no. 3; pp. 941 - 976
Main Authors Mardal, Kent-Andre, Rognes, Marie E., Thompson, Travis B.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.09.2021
BMJ Publishing Group
Subjects
Computational Mathematics and Numerical Analysis
Mathematics
Mathematics and Statistics
Numeric Computing
Stokes–Biot stability
74S05
Poroelasticity
Biot’s equations
Darcy stability
76S05
65M60
Mixed method
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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.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-021-00849-0