H1(X) and C0(X), respectively, the Bochner spaces of all L2, H1 and C0 functions
mapping the time interval [0,T] into X. For a measurable set Ω ⊆ Rd, we adopt
the simplified notation (·, ·)Ω and ∥·∥Ω for the scalar product and the norm in
L2(Ω). We write a ≲ b and a ≂ b when there are constants 0 < c ≤ c, possibly
different at different occurrences, such that a ≤ cb and cb ≤ a ≤ cb, respectively.
The hidden constants are independent of the material parameters involved in the
equations. The dependence on other relevant quantities is addressed case by case,
2. Inf-sup theory for the Biot’s equations
This section introduces the Biot’s equations and summarizes some results from
[24], that are useful for the construction and the analysis of the discretization in
the next sections.
2.1. Initial-boundary value problem. Let Ω ⊆ Rd, 1 ≤ d ≤ 3, be a bounded
domain with polyhedral and Lipschitz continuous boundary. The flow of a Newto-
nian fluid inside a linear elastic porous medium located in Ω, in the time interval
(0,T) with T > 0, is modeled by the quasi-static Biot’s equations as follows
(2.1a)
−div(2µ∇Su + (λdivu − αp)I) = fu
in Ω × (0,T)
∂t(αdivu + σp) − div(κ∇p) = fp
in Ω × (0,T).
The first equation states the momentum balance for the elastic porous medium,
whereas the second one is the mass balance for the fluid.
The unknown functions in the equations are the displacement u : Ω → Rd of the
elastic porous medium and the pressure p : Ω → R of the fluid. The differential
operator ∇S is the symmetric part of the gradient and I is d×d identity tensor. The
material parameters, denoted by Greek letters, are the Lamé constants µ,λ > 0,
the Biot-Willis constant α > 0, the constrained specific storage coefficient σ ≥ 0
and the hydraulic conductivity κ > 0. Consistently with [24], we assume that all
parameters are constant in Ω × [0,T] for simplicity. We refer to [24, Remark 2.1]
for a discussion on possible extensions.
We are interested in the initial-boundary value problem obtained by prescribing
also the initial condition
(2.1b)
(αdivu + σp)|t=0 = ℓ0
in Ω
as well as the boundary conditions
(2.1c)
u = 0
on Γu,E × (0,T)
(2µ∇Su + (λdivu − αp)I)n = gu
on Γu,N × (0,T)
p = 0
on Γp,E × (0,T)
κ∇p · n = gp
on Γp,N × (0,T)
where Γu,E ∪ Γu,N = ∂Ω=Γp,E ∪ Γp,N and Γu,E ∩ Γu,N = ∅ = Γp,E ∩ Γp,N . The
letter n denotes the outward unit normal vector on ∂Ω. The subscripts ‘E’ and
‘N’ are intended to assist the reader in distinguishing the portions of the boundary
with essential and natural conditions.
We point out that different statements of the initial and of the boundary con-
ditions are sometimes met in the literature. We refer to [24, Remark 2.2-2.3] for a
more extensive discussion on this point.