Inf-sup theory for the quasi-static Biot's equations in poroelasticity
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Date
2024-07
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Abstract
We analyze the two-field formulation of the quasi-static Biot's equations by means of the inf-sup theory. For this purpose, we exploit an equivalent four-field formulation of the equations, introducing the so-called
total pressure and total fluid content as independent variables. We establish existence, uniqueness and stability of the solution. Our stability estimate is two-sided and robust, meaning that the regularity established for the solution matches the regularity requirements for the data and the involved constants
are independent of all material parameters. We prove also that additional regularity in space of the data implies, in some cases, corresponding additional regularity in space of the solution. These results are instrumental to the design and the analysis of discretizations enjoying accurate stability and error
estimates.
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Keywords
poroelasticity, a posteriori analysis, well-posedness, Inf-sup theory