New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations

Abstract

In this paper we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the coupled poroelasticity and heat equations, which describe the interaction between the fields of deformation and temperature. A non-symmetric pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is an extension of Hooke's law to account for thermal effects. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each with right-hand terms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška-Brezzi theory in Banach spaces, allow to conclude, under a smallness assumption on the data, the existence of a unique solution. The discrete analysis is conducted in a similar manner, utilizing the Brouwer and Banach theorems to demonstrate both the existence and uniqueness of the discrete solution. The rates of convergence of the resulting Galerkin method are then presented. Finally, a number of numerical tests are shown to validate the aforementioned statement and demonstrate the good performance of the method.

Full text

UNIVERSIDAD DE CONCEPCI ´
ON
Centro de Investigaci´
on en
Ingenier
´
ıa Matem´
atica (CI2MA)
New Banach spaces-based mixed finite element methods for the
coupled poroelasticity and heat equations
Julio Careaga, Gabriel N. Gatica,
Cristian Inzunza, Ricardo Ruiz-Baier
PREPRINT 2024-02
SERIE DE PRE-PUBLICACIONES

New Banach spaces-based mixed finite element methods
for the coupled poroelasticity and heat equations∗
Julio Careaga†Gabriel N. Gatica‡Cristian Inzunza§Ricardo Ruiz-Baier¶
Abstract
In this paper we introduce and analyze a Banach spaces-based approach yielding a fully-mixed
finite element method for numerically solving the coupled poroelasticity and heat equations, which
describe the interaction between the fields of deformation and temperature. A non-symmetric
pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is
an extension of Hooke’s law to account for thermal effects. The resulting continuous formulation,
posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems,
each with right-hand terms that depend on data and the unknowns of the other two. The well-
posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem,
along with the Babuˇska–Brezzi theory in Banach spaces, allow to conclude, under a smallness
assumption on the data, the existence of a unique solution. The discrete analysis is conducted in
a similar manner, utilizing the Brouwer and Banach theorems to demonstrate both the existence
and uniqueness of the discrete solution. The rates of convergence of the resulting Galerkin method
are then presented. Finally, a number of numerical tests are shown to validate the aforementioned
statement and demonstrate the good performance of the method.
Key words: Thermo-poroelasticity, porous media, mixed finite element methods, analysis in Banach
spaces.
Mathematics subject classifications (2000): 65N30, 65J25, 74F05, 74F10.
1 Introduction
Scope. The relationship between the flow of a viscous fluid and the deformation of an elastic solid
within a porous medium is described by the poroelasticity equations, which were initially introduced
in the early works [34] and [7,8]. While porous materials are commonly associated with objects such
as rocks and clays, they also encompass a broader range of materials, including biological tissues,
foams, and even paper products. Moreover, in applications such as the underground disposal of
radioactive waste, geothermal energy production, and oil extraction from deep, high-temperature,
∗This work was partially supported by ANID-Chile through the projects Centro de Modelamiento Matem´
atico
(FB210005), Anillo of Computational Mathematics for Desalination Processes (ACT210087), and Fondecyt
postdoctoral project No. 3230553; by Centro de Investigaci´on en Ingenier´ıa Matem´atica (CI2MA); and by the Australian
Research Council through the Future Fellowship grant FT220100496 and Discovery Project grant DP220103160.
†Departamento de Matem´atica, Universidad del B´ıo B´ıo, Chile, email: jcareaga@ubiobio.cl.
‡CI2MA and Departamento de Ingenier´ıa Matem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile,
email: ggatica@ci2ma.udec.cl.
§CI2MA and Departamento de Ingenier´ıa Matem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile,
email: crinzunza@udec.cl.
¶School of Mathematics, Monash University, 9 Rainforest Walk, Melbourne VIC 3800, Australia; and Universidad
Adventista de Chile, Casilla 7-D, Chill´an, Chile, email: ricardo.ruizbaier@monash.edu.
1

high-pressure reservoirs, temperature plays a crucial role. Therefore, to study these phenomena, we
focus on the coupling between poroelasticity and heat equations. The resulting system, a slightly
modified version of the thermo-poroelastic problem [11,12,13], is non-linear and strongly coupled.
The set of equations consists of the steady-state balance of linear momentum for the mixture and
mass balance for the fluid content (using the modified Darcy law) and a convection-diffusion equation
depending on the Darcy seepage velocity and the total stress. In terms of numerical solvability,
a wide variety of techniques have been developed to simulate the poroelasticity problem, both by
itself [10] and when coupled with other equations. These include couplings with chemotaxis [4],
elasticity [2], Stokes [9,33] and diffusion [31]. The thermo-poroelasticity problem has also been recently
addressed in [11,12,13,35,36,37]. These references include primal formulations [35], a combination of
primal and mixed formulations [11,37], discontinuous Galerkin methods [3], a fully-mixed formulation
[12], and a mixed-primal-characteristics finite element method [37]. The introduction of additional
variables of physical relevance is a common approach to solving problems that involve couplings and
nonlinearities. Consequently, mixed methods are strongly justified in such a scenario. A recent
approach to this method consists of defining the corresponding variational formulation in terms of
Banach spaces instead of the usual Hilbertian framework without augmentation. It is important to
note that, although augmented methods allow the recovery of a Hilbertian framework, they increase
the cost of the computational implementation of the Galerkin scheme. Therefore, an analysis based
on Banach spaces has the advantage of studying the problem in its purest form. Another advantage of
this method lies in the relaxation of assumptions that must be made about the data, source terms, and
eventual solutions of the system. Consequently, the unknowns are now associated with the natural
spaces that result from the testing and integration by parts procedures; formulations of the models
become simpler and more faithful to the original physical models; momentum-conservative schemes
can be acquired; and additional unknowns can be calculated through postprocessing formulas. As a
non-exhaustive list of contributions taking advantage of the use of Banach frameworks for solving the
aforementioned kinds of problems, we refer to [14,16,17,20,23], and among the different models
considered there, we find elasticity, Brinkman–Forchheimer, Poisson–Nernst–Planck, Navier-Stokes,
chemotaxis/Navier–Stokes, Boussinesq, coupled flow-transport, and fluidized beds. For the coupled
poroelasticity and heat equations, however, no mixed methods with the aforementioned benefits have,
up to our knowledge, been developed yet. As motivated by the preceding discussion, the goal of this
paper is to develop a Banach spaces-based formulation leading to new mixed finite element methods
for the poroelasticity-heat model.
The manuscript is organized as follows. The rest of this section collects some preliminary notations,
definitions, and results to be utilized throughout the paper. In Section 2, we describe the model of
interest. In particular, we reformulate it in terms of the non-symmetric pseudostress tensor. In Section
3we derive the fully-mixed variational formulation of the problem by splitting the analysis according
to the three equations forming the coupled model. Suitable integration by parts formulae jointly with
the Cauchy–Schwarz and H¨older inequalities are crucial for determining the right Lebesgue and related
spaces to which the unknowns and corresponding test functions are required to belong. In Section
4, a fixed-point strategy is adopted to analyze the solvability of the continuous formulation. The
Babuˇska–Brezzi theory in Banach spaces is employed to study the corresponding uncoupled problems,
and then the classical Banach theorem is applied to conclude the existence of a unique solution. An
analog fixed-point approach to that of Section 4is utilized in Section 5to study the well-posedness of
the associated Galerkin scheme. Finally, numerical results showing how well the method works and
confirming the theoretical rates of convergence given in Section 5, are presented in Section 6.
Preliminaries. Throughout the paper Ω is a bounded Lipschitz-continuous domain of Rn,nP t2,3u,
which is star-shaped with respect to a ball, and whose outward unit normal at its boundary Γ is denoted
ν. Standard notation will be adopted for Lebesgue spaces LtpΩq, with tP r1,`8q, and Sobolev spaces
2

Wℓ,tpΩqand Wℓ,t
0pΩq, with ℓě0, whose corresponding norms and seminorms, either for the scalar,
vector, or tensorial version, are denoted by }¨}0,t;Ω ,}¨}ℓ,t;Ω, and |¨|ℓ,t;Ω, respectively. Note that
W0,tpΩq “ LtpΩq, and that when t“2, we simply write HℓpΩqinstead of Wℓ,2pΩq, with its norm and
seminorm denoted by }¨}ℓ;Ω and |¨|ℓ;Ω, respectively. Now, letting t, t1P p1,`8q conjugate to each
other, that is such that 1{t`1{t1“1, we let W1{t1,tpΓqand W´1{t1,t1pΓqbe the trace space of W1,tpΩq
and its dual, respectively, and denote the duality pairing between them by x¨,¨y. In particular, when
t“t1“2, we simply write H1{2pΓqand H´1{2pΓqinstead of W1{2,2pΓqand W´1{2,2pΓq, respectively.
Given any generic scalar functional space M, we let Mand Mbe its vector and tensorial counter-
parts. Furthermore, for any vector fields v“ pviqi“1,n and w“ pwiqi“1,n, we set the gradient and
divergence operators as
∇v:“ˆBvi
Bxj˙i,j“1,n
and divpvq:“
n
ÿ
j“1
Bvj
Bxj
.
In addition, for any tensor fields τ“pτijqi,j “1,n and ζ“pζij qi,j“1,n , we let divpτqbe the divergence
operator div acting along the rows of τ, and define the transpose, the trace, the tensor inner product,
and the deviatoric tensor, respectively, as
τt:“ pτji qi,j“1,n ,trpτq:“
n
ÿ
i“1
τii ,τ:ζ:“
n
ÿ
i,j“1
τij ζij ,τd:“τ´1
ntrpτqI,(1.1)
where Istands for the identity tensor of R:“Rnˆn. On the other hand, for each tP r1,`8q, we
introduce the Banach spaces
Hpdivt; Ωq:“!τPL2pΩq: divpτq P LtpΩq),
Htpdivt; Ωq:“!τPLtpΩq: divpτq P LtpΩq),
and
Htpdivt; Ωq:“!τPLtpΩq:divpτq P LtpΩq),
which are endowed with the natural norms
}τ}divt;Ω :“ }τ}0;Ω ` }divpτq}0,t;Ω @τPHpdivt; Ωq,
}τ}t,divt;Ω :“ }τ}0,t;Ω ` }divpτq}0,t;Ω @τPHtpdivt; Ωq,
and
}τ}t,divt;Ω :“ }τ}0,t;Ω ` }divpτq}0,t;Ω @τPHtpdivt; Ωq.
Then, we recall that, proceeding as in [25, eq. (1.43), Section 1.3.4] (see also [15, Section 4.1] and [21,
Section 3.1]), one can prove that for each tP#p1,`8s in R2,
r6
5,`8s in R3,there holds
xτ¨ν, vy “ żΩ!τ¨∇v`vdivpτq)@ pτ, vq P Hpdivt; Ωq ˆ H1pΩq,(1.2)
where x¨,¨y denotes the duality pairing between H1{2pΓqand H´1{2pΓq. In turn, given t, t1P p1,`8q
conjugate to each other, there also holds (cf. [24, Corollary B.57])
xτ¨ν, vy “ żΩ!τ¨∇v`vdivpτq)@ pτ, vq P Htpdivt; Ωq ˆ W1,t1pΩq,(1.3)
and analogously
xτ ν,vy “ żΩ!τ:∇v`v¨divpτq)@ pτ,vq P Htpdivt; Ωq ˆ W1,t1pΩq,(1.4)
where x¨,¨y denotes in (1.3) (resp. (1.4)) the duality pairing between W´1{t,tpΓq(resp. W´1{t,t pΓq)
and W1{t,t1pΓq(resp. W1{t,t1pΓq).
3

2 Governing equations and boundary conditions
We consider a homogeneous porous medium constituted by a mixture of incompressible grains and
interstitial fluid. The domain of interest Ω ĂRn, n “2,3, is assumed bounded. For a given body
force fand given source terms fand gneglecting convective, gravitational, and inertial terms, we
will concentrate the discussion on the following Biot’s equations coupled with a stationary convection-
diffusion equation modeling the heat of the mixture:
σ“2µepuq ` λdivpuqI´ pα p `β θqI,´divpσq “ fin Ω ,(2.1a)
χ p `αdivpuq ´ divpwq “ f , w“κ
η∇pin Ω ,(2.1b)
θ`w¨∇θ´divpDpσq∇θq “ gin Ω ,(2.1c)
u“uD, p “pDand θ“0 on Γ ,(2.1d)
where the tensor σis a generalized Hooke’s law, extended to include thermal effects, uis the unknown
vector of displacement of the solid particles, pis the bulk pressure of the fluid, wis the Darcy’s seepage
velocity and θis the temperature distribution. The remaining terms are the infinitesimal strain tensor
epuq:“1
2p∇u`∇utq, the permeability of the porous solid κ, the Lam´e constants of the solid (moduli
of dilation and shear, respectively) λand µ, the constrained specific storage coefficient χą0, the
Biot-Willis parameter αP p0,1s, the scaling of active stress that indicates a two-way coupling between
diffusion and motion β, the viscosity of the pore fluid η, and the stress-dependent diffusivity accounting
for an altered diffusion acting in the poroelastic domain D:RÑR.
Observe that tensor σis symmetric since epuqand Iare both symmetric. In order to avoid the weak
imposition of the symmetry of σ, we reformulate (2.1) in terms of the pseudostress ρ(non-symmetric
stress), defined by
ρ:“µ∇u` pµ`λqdivpuqI´ pα p `β θqIin Ω .(2.2)
Now, by applying trace to (2.2), we can express divpuqin terms of ρ,pand θ, namely
divpuq “ γpλq ptrpρq ` npαp `βθqq ,(2.3)
with the parameter-dependent coefficient
γpλq:“ pnλ ` pn`1qµq´1.(2.4)
While this coefficient depends also on µand n, only its dependence on λand its relation with other
model parameters will be important when we analyze the formulation in the quasi-incompressibility
limit. Replacing the obtained expression for divpuqinto (2.2) and using (1.1), we can equivalently
rewrite the equations in (2.1a) in terms of ρas follows
1
µρd`γpλq
ntrpρqI´∇u“ ´γpλq pα p `β θqI,´divpρq “ fin Ω .
Note that for the second equation above, we have used the fact that divpσq “ divpρq, which can be
corroborated by taking divergence to the first equation of (2.1a) and to (2.2), respectively. Moreover,
replacing (2.3) into the first equation of (2.1b), we obtain
c1pλqp´divpwq “ f´c2pλqtrpρq ´ c3pλqθ ,
where we have used the following parameter-dependent coefficients
c1pλq:“χ`n α2γpλq, c2pλq:“α γpλq,and c3pλq:“n α β γpλq.(2.5)
4

Again, we stress here the dependence on λonly. Next we reformulate (2.1c) in terms of ρwithin the
diffusivity function D, it is necessary to establish the function that maps σto the triple pρ, p, θq. In
this regard, from (2.1a), we have
2µepuq “ ρ`ρt´2pµ`λqdivpuqI`2pα p `β θqI,(2.6)
and thus, we deduce from (2.2), along with (2.3) and (2.6), that the original stress tensor σcan be
expressed in terms of the pseudostress ρ, pressure pand temperature θ, through the linear mapping
Cpρ, p, θq:“ρ`ρt´γpλq´p2µ`λqtrpρq`p2n´1pα p `β θq¯I“σ.(2.7)
Consequently, we can recast the original stress-dependent diffusivity Dby a function Kdepending on
ρ,pand θdefined by
Kpρ, p, θq:“DpCpρ, p, θqq .(2.8)
Finally, the model equations in (2.1) are restated, equivalently, on the unknowns ρ,pand θby the
coupled system:
1
µρd`γpλq
ntrpρqI´∇u“ ´γpλq pα p `β θqI,´divpρq “ fin Ω ,(2.9a)
c1pλqp´divpwq “ f´c2pλqtrpρq ´ c3pλqθ , η
κw´∇p“0in Ω ,(2.9b)
θ`w¨∇θ´div`Kpρ, p, θq∇θ˘“gin Ω ,(2.9c)
u“uD, p “pDand θ“0 on Γ .(2.9d)
Throughout this work, we suppose that, K:RˆRˆRÑRis a function of class C1and uniformly
positive definite, meaning the latter that there exists κ0ą0 such that
Kpτ, q, ξqs¨sąκ0|s|2,@ pτ, q, ξq P RˆRˆR.(2.10)
We also require uniform boundedness and Lipschitz continuity of K, that is that there exist positive
constants κ1,κ2and LK, such that
κ1ďKpτ, q, ξq ď κ2and |Kpτ, q, ξq ´ Kpτ0, q0, ξ0q| ď LK|pτ, q, ξq´pτ0, q0, ξ0q| ,(2.11)
for all pτ, q, ξq,pτ0, q0, ξ0q P RˆRˆR.It is pertinent to mention here that one of the main consequences
of introducing the new variable ρis that (2.9c) becomes nonlinear with respect to θunlike (2.1c).
Furthermore, it is easily seen from (2.7) and (2.8) that sufficient conditions for (2.11) are given by
analogue conditions for D, that is by the existence of positive constants δ1,δ2, and LD, such that
δ1ďDpτq ď δ2and |Dpζq ´ Dpτq| ď LD|ζ´τ| @ ζ,τPR.
3 Mixed weak formulation
In this section, we derive a mixed formulation of the system (2.9). To this end, we treat each variational
formulation of (2.9a), (2.9b) and (2.9c) independently, ending up with three systems whose coupling
is carried out via a fixed-point iteration strategy.
5

3.1 Mixed formulation of the poroelasticity equations
In what follows, we are going to address the mixed formulation for the poroelasticity equations in
(2.9a) for a given pressure pand temperature θ, which are going to be determined by (2.9b) and
(2.9c), respectively. The poroelasticity equations defined for the non-symmetric pseudostress ρand
velocity uunknowns are given by
1
µρd`1
nγpλqtrpρqI´∇u“ ´ γpλq pα p `β θqIin Ω ,
´divpρq “ fin Ω ,and u“uDon Γ .
(3.1)
We notice that in order to properly couple the equations (2.9), we need to be able to control the
following expression associated with the heat equation
żΩ`Kpζ,rp, ϑq ´ Kpζ0,rp0, ϑ0q˘t¨s,
where pζ,r
p, ϑqand pζ0,r
p0, ϑ0qbelong to the same space in which we will seek the unknowns pρ, p, θq,
and the functions tand sare generic vectors that belong to the same space than ∇θ. In this regard,
and employing the Lipschitz-continuity property of K(cf. (2.11)), straightforward applications of
Cauchy–Schwarz and H¨older inequalities yield
ˇˇˇˇżΩ`Kpζ,rp, ϑq ´ Kpζ0,rp0, ϑ0q˘t¨sˇˇˇˇ
ďLK´}ζ´ζ0}0,2j;Ω ` }rp´rp0}0,2j;Ω ` }ϑ´ϑ0}0,2j;Ω ¯}t}0,2k;Ω }s}0;Ω ,
(3.2)
where j,kP p1,`8q are conjugate to each other. The latter inequality makes sense for ζ,ζ0PL2jpΩq,
r
p, rp0, ϑ and ϑ0PL2jpΩq, and tPL2kpΩq. In this way, the above leads us to initially look for ρin the
space LrpΩq,pPLrpΩqand θinitially in LrpΩq, with r:“2j. The specific choice of rwill be discussed
later on, so that meanwhile we consider a generic rand let sP p1,2qbe its respective conjugate. In
turn, a suitable bounding of }t}0,2k;Ω in (3.2) for a particular twill also be explained subsequently by
means of a regularity argument.
With the preliminary choice of the space to which ρbelongs established above, it follows now from
the first equation of (3.1) that ushould be initially sought in W1,rpΩq. Thus, in order to derive the
variational formulation for the poroelasticity equations, we need to invoke a suitable integration by
parts formula. Indeed, applying (1.4) with t“sand t1“rto uPW1,rpΩq, for which we assume from
now on that uDbelongs to W1{s,rpΓq, we find that
żΩ
∇u¨τ“ ´ żΩ
u¨divpτq`xτ ν,uDyΓ,
so that, the testing of the first equation of (3.1) against τPHspdivs; Ωqgives
1
µżΩ
ρd:τd`γpλq
nżΩ
trpρqtrpτq ` żΩ
u¨divpτq“xτ ν ,uDyΓ´γpλqżΩ
pα p `β θqtrpτq.(3.3)
Here, we notice that the second term on the right-hand side of (3.3) does indeed make sense for pand
θinitially in LrpΩq. In fact, thanks to H¨older’s inequality we have
żΩ
ptrpτq ď n1{r}p}0,r;Ω }τ}0,s;Ω ,żΩ
θtrpτq ď n1{r}θ}0,r;Ω }τ}0,s;Ω .(3.4)
6

As a result, the third term on the left-hand side of (3.3) implies that it is sufficient to consider uin
LrpΩq. Additionally, when testing the second equation of (3.1) against vPLspΩq, we obtain
żΩ
v¨divpρq“´żΩ
f¨v,(3.5)
which makes sense when divpρq P LrpΩqand fPLrpΩq, the latter being assumed in what follows, and
thus from now on we seek ρin Hrpdivr; Ωq. In addition, we notice that for each tP p1,`8q there
holds the decomposition
Htpdivt; Ωq “ Ht
0pdivt; Ωq ‘ RI,with Ht
0pdivt; Ωq:“!τPHtpdivt; Ωq:żΩ
trpτq “ 0).(3.6)
Note that replacing τby the identity tensor Iin (3.3) and using that the deviator of Iis the null
tensor, we get an expression for the integral of the trace of ρ, this is
żΩ
trpρq “ 1
γpλqżΓ
u¨ν´nżΩ
pαp `βθq.(3.7)
Now, using the decomposition (3.6) with t“r, we have that ρ“ρ0`cIwith unique ρ0PHr
0pdivt; Ωq
and constant cPR, which thanks to (3.7), can be computed by
c:“1
n|Ω|żΩ
trpρq “ 1
nγpλq|Ω|żΓ
uD¨ν´1
|Ω|żΩ
pα p `β θq.(3.8)
Hence, ccan be obtained once the pressure and temperature are known, and in order to fully attain
the explicit knowledge of the unknown ρ, it only remains to find its Hr
0pdivr; Ωq-component ρ0. On
the other hand, (3.6) also applies to each τin Hspdivs; Ωqwith unique decomposition τ“τ0`dI,
for τ0PHs
0pdivs; Ωqand respective constant dPR.
Therefore, we reformulate our problem in terms of ρ0instead. To do so, we replace ρ“ρ0`cI
into (3.3) and (2.9b), denote ρ0simply by ρand substitute Kpρ, p, θqby Kpρ`cI, p, θqin the heat
equation (2.9c). Furthermore, we observe that testing (3.3) against τPHspdivs; Ωqis equivalent to
doing it against τPHs
0pdivs; Ωq, which together with the above, leads us to consider the following
Banach spaces
X2:“Hr
0pdivr; Ωq,M1:“LrpΩq,X1:“Hs
0pdivs; Ωq,M2:“LspΩq,
so that, given p, θ PLrpΩq, and gathering (3.3) and (3.5), we arrive at the following mixed formulation
for the poroelasticity equations (2.9a): Find pρ,uq P X2ˆM1such that
apρ,τq ` b1pτ,uq “ Fp,θpτq @ τPX1,
b2pρ,vq “ Gpvq @ vPM2,
(3.9)
where the bilinear forms a:X2ˆX1ÑR and bi:XiˆMiÑR, with iP t1,2u, are defined by
apρ,τq:“1
µżΩ
ρd:τd`γpλq
nżΩ
trpρqtrpτq @ pρ,τq P X2ˆX1,
bipτ,vq:“żΩ
v¨divpτq @ pτ,vq P XiˆMi.
In turn, given q, ϑ in LrpΩq, the linear functionals Fq,ϑ :X1ÑR and G:M2ÑR, are defined by
Fq,ϑpτq:“ xτ ν,uDyΓ´γpλqżΩ
pα q `β ϑqtrpτq @ τPX1,(3.10a)
Gpvq:“ ´ żΩ
f¨v@vPM2.(3.10b)
7

Next, it is easily seen that a,b1,b2and Gare bounded. In fact, applying H¨older’s inequality, we find
that there exist positive constants, denoted and given by
}a}:“2
µ,}bi}:“1 and }G} “ }f}0,r ;Ω ,(3.11)
such that |apρ,τq| ď }a} }ρ}X2}τ}X1@ pρ,τq P X2ˆX1,
|bipτ,vq| ď }b} }τ}Xi}v}Mi@ pτ,vq P XiˆMi,
|Gpvq| ď }G} }v}M2@vPM2.
Regarding the boundedness of the functional Fp,θ , where pand θare initially in LrpΩq, we will establish
this in the forthcoming Section 3.3, where the range of rwill be determined for each unknown.
3.2 Mixed formulation of the perturbed Darcy problem
Continuing with the weak formulation of (2.9), we are going to focus now on the perturbed Darcy
equation (2.9b) including the boundary condition of the pressure, for a given ρPX2and p, θ PLrpΩq.
Following the derivation done in Section 3.1, we use decomposition (3.6) together with the definition
of c(cf. (3.8)), and replace trpρqby trpρ`cIqinto (2.9b), so that the perturbed Darcy problem
describing the velocity wand pressure pis then given by
η
κw´∇p“0in Ω ,
divpwq ´ c1pλqp“c2pλqtrpρ`cIq ` c3pλqθ´fin Ω ,
p“pDon Γ ,
(3.12)
where the constant cmultiplying Ion the right-hand side of the second equation is defined by (3.8),
and depends on pand θ. Next, given tP p1,8q, we consider the zero mean mapping m: LtpΩq Ñ Lt
0pΩq
defined by
mpqq:“q´1
|Ω|żΩ
q@qPLtpΩq.(3.13)
Then, replacing (3.8) and using the notation q0:“mpqq P Lt
0pΩq, the second equation of (3.12) can
be written as
divpwq ´ χ p ´n α2γpλqp0“c2pλqtrpρq ` c3pλqθ0`α
|Ω|żΓ
uD¨ν´f . (3.14)
Prior to addressing the weak formulation of (3.12), we notice that in order to properly couple (3.14)
to equation (2.9c), we need to be able to control the expression
żΩ
pw¨∇θqϑ ,
which arises later on (cf. (3.25)) when dealing with the variational formulation of the heat equation.
Here ϑis a function belonging to the same space in which we will seek the temperature θ. Applying
generalized H¨older’s inequality to the triple product present in the above integral, we get
ˇˇˇˇżΩ
pw¨∇θqϑˇˇˇˇď }w}0,2j;Ω }∇θ}0;Ω }ϑ}0,2k;Ω ,(3.15)
where j, k P p1,`8q are conjugate to each other, and the inequality holds true for wPLrpΩq,
∇θPL2pΩq, and ϑPLρpΩq, with pr, ρq:“ p2j, 2kq. Considering that θis initially taken from LrpΩq,
8

we have to require that rďρ, a condition that will be satisfied when determining the range for ρ, so
that for now we consider ρP p2,`8q, and let ϱbe its respective conjugate.
Having chosen LrpΩqas the preliminary space for w, (3.12) tentatively suggests to look for pin
W1,rpΩq. In this way, testing the first equation of (3.12) against zPHspdivs; Ωq, and applying (1.3)
together with the Dirichlet boundary condition for p, we obtain
η
κżΩ
w¨z`żΩ
pdivpzq “ xz¨ν, pDyΓ@zPHspdivs; Ωq,(3.16)
which requires to assume that pDPW1{s,rpΓq. Then, a straightforward application of H¨older’s in-
equality in the second term on the left-hand side of (3.16) shows that it suffices to seek the pressure
pin the space LrpΩq, which coincides with the space obtained in (3.4). On the other hand, testing
(3.14) against an arbitrary function qbelonging to a space to be determined, we formally get
żΩ
qdivpwq ´ χżΩ
p q ´nα2γpλqżΩ
p0q
“c2pλqżΩ
qtrpρq ` c3pλqżΩ
θ0q`α
n|Ω|żΓ
uD¨νżΩ
q´żΩ
f q .
(3.17)
Since we will look for pin LrpΩq, a direct application of the H¨older’s inequality implies that the
second term on the left-hand side of (3.17) makes sense if qis considered in LspΩq. Consequently, the
remaining terms of (3.17) make sense if divpwqand fbelong to LrpΩq, and then wmust be sought
in Hrpdivr; Ωq. In this way, we define the following spaces
X2:“Hrpdivr; Ωq,X1:“Hspdivs; Ωq,M1:“LrpΩqand M2:“LspΩq.(3.18)
Then, given pρ, θq P X2ˆLρpΩq, the mixed formulation for the perturbed Darcy equation reduces to:
Find pw, pq P X2ˆM1such that
cpw,zq ` d1pz, pq “ Fpzq @ zPX1,
d2pw, qq ´ epp, qq “ Gρ,θ pqq @ qPM2,
(3.19)
where the bilinear forms c:X2ˆX1ÑR, di:XiˆMiÑR, iP t1,2u, and e: M1ˆM2ÑR, which
are independent of ρand θ, are defined by
cpw,zq:“η
κżΩ
w¨z@ pw,zq P X2ˆX1,(3.20a)
dipz, qq:“żΩ
qdivpzq @ pz, qq P XiˆMi,(3.20b)
and
epp, qq:“χżΩ
p q `n α2γpλqżΩ
p0q@ pp, qq P M1ˆM2.(3.20c)
Furthermore, the functionals F:X1ÑR and Gζ,ϑ : M2ÑR, for each pζ, ϑq P X2ˆLρpΩq, are
defined by
Fpzq:“ xz¨ν, pDyΓ@zPX1,and (3.21a)
Gζ,ϑpqq:“c2pλqżΩ
qtrpζq ` c3pλqżΩ
ϑ0q`α
n|Ω|żΓ
uD¨νżΩ
q´żΩ
fq @qPM2.(3.21b)
In addition, the bilinear forms c,di,iP t1,2uand eare all bounded. Finally, applying Cauchy–
Schwarz and H¨older inequalities, we find that there exist positive constants, given by
}c}:“η
κ,}di}:“1,}e}:“max ␣χ , n α2γpλq(,(3.22)
9

such that
|cpw,zq| ď }c} }w}X2}z}X1@ pw,zq P X2ˆX1,
|dipz, qq| ď }di} }z}Xi}q}Mi@ pz, qq P XiˆMi, i P t1,2u,
|epp, qq| ď }e} }p}M1}q}M2@ pp, qq P M1ˆM2.
The boundedness of Fand Gζ,ϑ will be proven later in the next section.
3.3 Mixed formulation of the heat equation
We treat now the mixed formulation of (2.9c) for a given ρPX2and wPX2. For this purpose,
we define two auxiliary unknowns, the gradient of the temperature and the term contained in the
argument of the divergence operator in (2.9c), this is
r
t:“∇θand r
σ:“Kpρ, p, θqr
t.(3.23)
Then, replacing these variables, the heat equation (2.9c) describing the temperature θcan be written
as r
t“∇θ, r
σ“Kpρ, p, θqr
tand θ`w¨r
t´divpr
σq “ gin Ω ,
θ“0 on Γ .
(3.24)
Now, testing the third equation of (3.24) against an arbitrary function ϑbelonging to a space to be
determined, we formally get
żΩ
θ ϑ `żΩ
w¨r
tϑ´żΩ
ϑdivpr
σq “ żΩ
g ϑ . (3.25)
Next, proceeding as in (3.15), we notice that applying generalized H¨older’s inequality to the triple
product in the second term on the left-hand side of (3.25) we get
ˇˇˇˇżΩ
w¨r
tϑˇˇˇˇď }w}0,r;Ω }r
t}0;Ω }ϑ}0,ρ;Ω ,
whence we can look for r
tPL2pΩqand ϑPLρpΩq. In addition, performing similar calculations as
before but over the first term on the left-hand side of (3.25), for ρ“2ką2, we obtain
ˇˇˇˇżΩ
θ ϑˇˇˇˇď }θ}0;Ω }ϑ}0;Ω ď |Ω|ρ´2
ρ}θ}0,ρ;Ω }ϑ}0,ρ;Ω ,
and in consequence θcan be sought in the same space as ϑ, its associated test function, which is
LρpΩq. In light of this, the data gwill be considered in LϱpΩq. Furthermore, a direct application of
H¨older’s inequality yields the third term on the left-hand side of (3.25) to be bounded as follows
ˇˇˇˇżΩ
ϑdivpr
σqˇˇˇˇď }ϑ}0,ρ;Ω }divpr
σq}0,ϱ;Ω ,
where, recalling that ϱis the conjugate of ρ, we observe that this term makes sense as long as
divpr
σq P LϱpΩq. Moreover, since r
tPL2pΩqand Kis bounded (cf. (2.11)), we can test the second
equation of (3.23) against r
sin L2pΩq, that is
´żΩr
σ¨r
s`żΩ
Kpρ, p, θqr
t¨r
s“0@r
sPL2pΩq,(3.26)
10

where, from the first term, we obtain that r
σmust be searched in L2pΩq, and more specifically in
Hpdivϱ; Ωqaccording to the preceding discussion.
Now, we observe that from the first equation of (3.24) we need θPH1pΩq, but since θPLρpΩq
this condition will be valid if H1pΩqis continuously embedded in LρpΩq. The latter is guaranteed for
ρP r1,`8q when n“2, which is always satisfied in the two-dimensional case, and for ρP r1,6swhen
n“3. Furthermore, in order to prove an inf-sup condition associated to wwe are going to apply,
e.g. [27, Theorem 3.2], which requires that rP r4{3,4swhen n“2 and rP r3{2,3swhen n“3. On
the other hand, since rą2 (see Section 3.1), the respective lower bounds are already satisfied, and we
only need to verify the upper ones. We readily observe that since r“2ρ{pρ´2q, for n“2, rď4 if
only if ρě4, whereas for n“3, rď3 if only if ρě6. Thus, intersecting the above with the previous
restrictions on ρ, we find that when n“2 we require ρě4, and when n“3 the only possible choice
is ρ“6. Therefore, we conclude that the feasible ranges for pr, ρqand their respective conjugates,
ps, ϱq, are given by
#rP p2,4sand sP r4{3,2qif n“2,
r“3 and s“3{2 if n“3,#ρP r4,`8q and ϱP p1,4{3sif n“2,
ρ“6 and ϱ“6{5 if n“3.(3.27)
Then, bearing in mind that r
tand θbelong to L2pΩqand LρpΩq, respectively, we test the first equation
of (3.23) against a r
τPHpdivϱ; Ωqand applying (1.2), we formally get
żΩr
t¨r
τ`żΩ
θdivpr
τq “ 0@r
τPHpdivϱ; Ωq.(3.28)
Consequently, taking into account the foregoing discussion, we introduce the following spaces and
notation to be used in our formulation:
H1:“LρpΩq,H2:“L2pΩq,H:“H1ˆH2,Q:“Hpdivϱ; Ωq,

θ:“ pθ, r
tq,
ϑ:“ pϑ, r
sq P H.
Finally, suitably gathering (3.25), (3.26) and (3.28), for a given 
p:“ pρ,w, pq P X2ˆX2ˆM1, we
arrive at the following mixed formulation for the heat equation: Find p
θ, r
σq:“`pθ, r
tq,r
σ˘PHˆQ
such that
a
p,θp
θ, 
ϑq ` bp
ϑ, r
σq “ Fp
ϑq @ 
ϑ:“ pϑ, r
sq P H,
bp
θ, r
τq “ 0@r
τPQ,(3.29)
where, given 
q“ pζ,z, qq P X2ˆX2ˆM1and ξPH1,a
q,ξ :HˆHÑR and b:HˆQÑR are the
bilinear forms defined by
a
q,ξp
θ, 
ϑq:“żΩ
θ ϑ `żΩ
K`ζ, q, ξ˘r
t¨r
s`żΩ
z¨r
tϑ@
θ, 
ϑPH,(3.30a)
bp
ϑ, r
τq:“ ´ żΩr
τ¨r
s´żΩ
ϑdivpr
τq @ p
ϑ, r
τq P HˆQ.(3.30b)
It is important to notice that, since a
p,θ involves the function Kin its definition, which in turn depends
on θ, the term a
p,θp
θ, 
ϑqis nonlinear. Additionally, the functional F:HÑR is given by
Fp
ϑq:“żΩ
gϑ @
ϑ“ pϑ, r
sq P H.
Next, it is easily seen that, given 
qPX2ˆX2ˆM1and ξPH1,a
q,ξ,b, and Fare bounded. In fact,
endowing Hand Qwith the norms
}
ϑ}H:“ }ϑ}0,ρ;Ω ` }r
s}0;Ω @
ϑPH,}r
τ}Q:“ }r
τ}divϱ;Ω @r
τPQ,
11

and applying the Cauchy–Schwarz and H¨older inequalities, we find that there exist positive constants,
denoted and given by
}a}:“maxt|Ω|pρ´2q{2ρ,κ2u,}b}:“1,and }F} “ }g}0,ϱ;Ω ,(3.31)
such that
|a
q,ξp
θ, 
ϑq| ď `}a}`}z}0,r;Ω ˘}
θ}H}
ϑ}H@
θ, 
ϑPH,
|bp
ϑ, r
σq| ď }b} }
ϑ}H}r
τ}Q@ p
ϑ, r
τq P HˆQ,
|Fp
ϑq| ď }F} }
ϑ}H@
ϑPH.
(3.32)
Regarding the boundedness of Fq,ϑ ,Fand Gζ,ϑ (cf. (3.10a),(3.21a) and (3.21b), respectively), we
observe that knowing already that pq, ϑq P LrpΩq ˆ LρpΩq, with rand ρwithin the ranges stipulated
by (3.27), invoking the identity (1.3), the continuous injections ir:H1pΩq Ñ LrpΩqand iρ: H1pΩq Ñ
LρpΩq, the definitions of the constants c2pλqand c3pλq(cf. (2.5)), and employing the Cauchy–Schwarz
and H¨older inequalities, we can conclude that there exist positive constants CF,CF, and CG, depending
on n,r,ρ,}ir},}iρ},|Ω|,αand β, so that letting
}Fq,ϑ}:“CF!}uD}1{s,r;Γ `γpλq´}q}0,r;Ω ` }ϑ}0,ρ;Ω¯),
}F}:“CF}pD}1{s,r;Γ ,and
}Gζ,ϑ}:“CG!}f}0,r;Ω ` }uD}1{s,r ;Γ `γpλq´}ζ}X2` }ϑ}0,ρ;Ω¯),
there holds |Fq,ϑpτq| ď }Fq,ϑ} }τ}X1@τPX1,
|Fpzq| ď }F} }z}X1@zPX1,and
|Gζ,ϑpqq| ď }Gζ,ϑ } }q}M2@qPM2.
(3.33)
3.4 The coupled fully-mixed formulation
Following the derivations presented in the previous sections, the fully-mixed formulation for (2.9)
reduces to gathering (3.9), (3.19) and (3.29), that is: Find pρ,uq P X2ˆM1,pw, pq P X2ˆM1and
p
θ, r
σq:“`pθ, r
tq,r
σ˘PHˆQsuch that
apρ,τq ` b1pτ,uq “ Fp,θpτq @ τPX1,
b2pρ,vq “ Gpvq @ vPM2,
cpw,zq ` d1pz, pq “ Fpzq @ zPX1,
d2pw, qq ´ epp, qq “ Gρ,θ pqq @ qPM2,
a
p,θp
θ, 
ϑq ` bp
ϑ, r
σq “ Fp
ϑq @ 
ϑPH,
bp
θ, r
τq “ 0@r
τPQ,
(3.34)
where 
p“ pρ,w, pq P X2ˆX2ˆM1.
4 The continuous solvability analysis
In this section, we will first use the Babuˇska–Brezzi theory in Banach spaces (cf. [6, Theorem 2.1,
Corollary 2.1, Section 2.1] for the general case, and [24, Theorem 2.34] for a particular one) to address
12

the well-posedness of each one of the decoupled problems arising from (3.9), (3.19), and (3.29). Then,
we proceed similarly as in [21] and [29] (see also [15], [30], and some references therein), and adopt a
fixed-point strategy to analyze the solvability of the fully coupled system (3.34).
4.1 The decoupled poroelasticity equations
We begin by introducing the operator S: M1ˆH1ÑX2defined by
Spq, ϑq:“ρ@ pq, ϑq P M1ˆH1,
where pρ,uq P X2ˆM1is the unique solution (to be confirmed below) of the mixed formulation arising
from (3.9) after replacing pp, θqby pq, ϑq, that is
apρ,τq ` b1pτ,uq “ Fq,ϑpτq @ τPX1,
b2pρ,vq “ Gpvq @ vPM2.
(4.1)
In order to prove that (4.1) is well-posed (equivalently, that Sis well-defined), we notice that (4.1) has
the same bilinear forms of [28, eq. (3.15)]. Then, assuming that the Lam´e parameter is sufficiently
large, namely λąM, where Mis specified in [28, Lemma 3.4], we can establish that the operator
Sis well defined. Indeed, letting αA,β1, and β2be the constants yielding the continuous inf-sup
conditions for a,b1, and b2(cf. [28, Lemmas 3.4 and 3.5]), we have the following result.
Lemma 4.1. Let rand sbe within the range of values stipulated by (3.27), and assume that λąM.
Then, for each pq, ϑq P M1ˆH1there exists a unique pρ,uq P X2ˆM1solution of (4.1), and hence
one can define Spq, ϑq:“ρ. Moreover, there exists a positive constant CS, depending on αA,β1,β2,
CF, and µ, such that
}Spq, ϑq} “ }ρ}X2ďCS"}uD}1{s,r;Γ ` }f}0,r;Ω `γpλq´}q}0,r;Ω ` }ϑ}0,ρ;Ω¯*.(4.2)
Proof. Thanks to the fact that Xiand Mi, with i“ t1,2u, are reflexive Banach spaces, along with
the boundedness of all the forms and functionals involved, and the inf-sup conditions provided by [28,
Lemmas 3.4 and 3.5], the proof reduces to a direct application of [6, Theorem 2.1, Corollary 2.1].
In particular, the a priori estimate (4.2) follows from [6, Corollary 2.1, eq. (2.15)]. Note that the
dependence of the constant CSon µis due to }a}(cf. (3.11)).
Regarding the a priori estimate for the component uPM1of the unique solution of (4.1), we recall
that, given pq, ϑq P M1ˆH1, the second inequality in [6, Corollary 2.1] yields
}u}M1ď¯
CS"}uD}1{s,r;Γ ` }f}0,r;Ω `γpλq´}q}0,r;Ω ` }ϑ}0,ρ;Ω ¯*,
where ¯
CSis a positive constant which depends principally on CF,αA,β1and β2.
4.2 The decoupled perturbed Darcy problem
As in Section 4.1, we now introduce the operator Ξ:X2ˆH1ÑX2ˆM1defined by
Ξpζ, ϑq “ `Ξ1pζ, ϑq,Ξ2pζ, ϑq˘:“ pw, pq @ pζ, ϑq P X2ˆLρpΩq,
13

where pw, pq P X2ˆM1is the unique solution (to be confirmed below) of the mixed formulation arising
from (3.19) after replacing pρ, θqby pζ, ϑq, that is
cpw,zq ` d1pz, pq “ Fpzq @ zPX1,
d2pw, qq ´ epp, qq “ Gζ,ϑpqq @ qPM2.
(4.3)
We observe that (4.3) has a perturbed saddle point structure over Banach spaces, but the fact that
the trial and test spaces are different prevent us from using, e.g. [22, Theorem 3.1], and therefore
an additional treatment is needed. Then, proceeding as in [20, Section 3.2.3], we first employ the
Babˇska–Brezzi theory in Banach spaces (cf. [6, Theorem 2.1, Corollary 2.1, Section 2.1]) to analyze
part of (4.3), and then apply the Banach–Neˇcas–Babuˇska theorem (cf. [24, Theorem 2.6]) to conclude
the well-posedness of the whole problem. According to this, we now let r
A:pX2ˆM1qˆpX1ˆM2q Ñ R
be the bounded bilinear form arising from (4.3) after adding the left-hand sides of its equations, but
without including e, that is
r
Appw, pq,pz, qqq :“cpw,zq ` d1pz, pq ` d2pw, qq @ pw, pq P X2ˆM1@ pz, qq P X1ˆM2,(4.4)
and aim to prove next that r
Asatisfies global continuous inf-sup conditions with respect to both its first
and second component. Note that the boundedness of r
Afollows from those of c,d1and d2(cf.(3.20a)
and (3.20b)). The verification of the aforementioned properties of r
Ais equivalent to establishing that
the bilinear forms c,d1and d2verify the hypotheses of [6, Theorem 2.1], which we address in what
follows. Firstly, according to the definitions of Xiand Mi(cf. (3.18)), the kernel of the operators di,
iP t1,2u, are given by
V1:“!zPHspdivs; Ωq: divpzq “ 0)and V2:“!zPHrpdivr; Ωq: divpzq “ 0).
The two subsequent lemmas, akin to those previously stated and demonstrated in [20] and [29],
establish the inf-sup conditions required by [6, Theorem 2.1] for the bilinear forms c(cf. (3.20a)), and
d1,d2(cf. (3.20b)), respectively.
Lemma 4.2. Assume that rand ssatisfy the particular range specified by (3.27). Then, there exists
a positive constant αcsuch that
sup
zPV1
z‰0
cpw,zq
}z}X1
ěαc}w}X2@wPV2,
and
sup
wPV2
cpw,zq ą 0@zPX1,z‰0.
Proof. The proof follows a similar approach as in [20, Lemma 3.4], leading to αc“η
κ}Ds}, with Ds
being the bounded linear operator introduced in [20, Lemma 3.3].
The continuous inf-sup conditions for the bilinear forms di,iP t1,2uare presented next.
Lemma 4.3. For each iP t1,2uthere exists a positive constant r
βisuch that
sup
zPXi
z‰0
dipz, qq
}z}Xi
ěr
βi}q}Mi@qPMi.
14

Proof. A proof of this lemma can be done by slightly modifying that of [29, Lemma 2.7], considering
Dirichlet boundary conditions of the auxiliary problems instead.
According to Lemmas 4.2 and 4.3, the required hypotheses of [6, Theorem 2.1, Section 2.1] are
satisfied, and hence the a priori estimation provided by [6, Corollary 2.1, Section 2.1] imply the
existence of a positive constant αA, depending only on αc,r
β1,r
β2and }c}, such that
sup
pz,qqPX1ˆM2
pz,qq‰0r
Appw, pq,pz, qqq
}pz, qq}X1ˆM2
ěαA}pw, pq}X2ˆM1@ pw, pq P X2ˆM1,(4.5a)
sup
pw,pqPX2ˆM1
pw,pq‰0r
Appw, pq,pz, qqq
}pw, pq}X2ˆM1
ěαA}pz, qq}X1ˆM2@ pz, qq P X1ˆM2.(4.5b)
Therefore, we let A:pX2ˆM1q ˆ pX1ˆM2q Ñ R be the bounded and linear operator arising from
(4.3) after adding the full left-hand sides of its equations, that is
Appw, pq,pz, qqq “ cpw,zq ` d1pz, pq ` d2pw, qq ´ epp, qq
@ pw, pq P X2ˆM1,@ pz, qq P X1ˆM2.
(4.6)
Having introduced this operator, we realize that solving (4.3) for a given pair pζ, ϑq P X2ˆH1, is
equivalent to: Find pw, pq P X2ˆM1such that
Appw, pq,pz, qqq “ Fpzq ` Gζ,ϑpqq @ pz, qq P X1ˆM2.
We notice that, thanks to the boundedness of r
Aand e, the operator Ais bounded as well. Thus
bearing in mind (4.6), employing (4.5a) and the boundedness of e(cf. (3.22)), we have
sup
pz,qqPX1ˆM2
pz,qq‰0
Appw, pq,pz, qqq
}pz, qq}X1ˆM2
ě!αA´ }e})}pw, pq}X2ˆM1@ pw, pq P X2ˆM1.
Then, assuming that the data satisfy
}e}:“max ␣χ , n α2γpλq(ďαA
2,(4.7)
we arrive at the global inf-sup condition for the perturbed Darcy problem
sup
pz,qqPX1ˆM2
pz,qq‰0
Appw, pq,pz, qqq
}pz, qq}X1ˆM2
ěαA
2}pw, pq}X2ˆM1@ pw, pq P X2ˆM1.(4.8)
Similarly, but employing now (4.5b) instead of (4.5a), and under the same assumption (4.7), we obtain
the second desired inf-sup condition for A, this is
sup
pw,pqPX2ˆM1
pw,pq‰0
Appw, pq,pz, qqq
}pw, pq}X2ˆM1
ěαA
2}pz, qq}X1ˆM2@ pz, qq P X1ˆM2.(4.9)
We are now in position to establish the well-posedness of the operator Ξ, equivalently the existence
and uniqueness of solution of (4.3).
15

Lemma 4.4. Let rand sbe within the range of values stipulated by (3.27), and assume that the data
fulfill condition (4.7). Then, for each pζ, ϑq P X2ˆH1there exists a unique pw, pq P X2ˆM1solution
of (4.3), and hence one can define Ξpζ, ϑq:“ pw, pq P X2ˆM1. Moreover, there exists a positive
constant CΞ, depending on αA,CF, and CG, such that
}Ξpζ, ϑq}X2ˆM1“ }w}X2` }p}M1
ďCΞ"}pD}1{s,r;Γ ` }f}0,r;Ω ` }uD}1{s,r;Γ `γpλq´}ζ}X2` }ϑ}0,ρ;Ω ¯*.(4.10)
Proof. Given pζ, ϑq P X2ˆH1, thanks to the boundedness of A, and the global inf-sup conditions
(4.8) and (4.9), a direct application of [24, Theorem 2.6] provides the existence of a unique solution
pw, pq P X2ˆM1to (4.3). The a priori estimate (cf. [24, Theorem 2.6, eq. (2.5)]) yields
}Ξpζ, ϑq}X2ˆM1“ }w}X2` }p}M1ď2
αA!}F}`}Gζ,ϑ }),
which, together with the expressions for }F},}Gζ,ϑ}given in (3.33) imply (4.10).
4.3 The decoupled heat equation
We now introduce the operator Π:X2ˆ pX2ˆM1q ˆ H1ÑHdefined by
Πpζ,
z, ξq “ `Π1pζ,
z, ξq,Π2pζ,
z, ξq˘:“
θ“ pθ, r
tq,
for all pζ,
z, ξq “ `ζ,pz, qq, ξ˘PX2ˆ pX2ˆM1q ˆ H1, where p
θ, r
σq “ `pθ, r
tq,r
σ˘PHˆQis the
unique solution (to be confirmed below) of the problem arising from (3.29) after replacing a
p,θ, with

p“ pρ,ω, pq, by a
q,ξ, with 
q“ pζ,z, qq, that is
a
q,ξp
θ, 
ϑq ` bp
ϑ, r
σq “ Fp
ϑq @ 
ϑ:“ pϑ, r
sq P H,
bp
θ, r
τq “ 0@r
τPQ.(4.11)
We recall from (3.32) that the bilinear form a
q,ξ (cf. (3.30a)) is bounded with constant }a} ` }z}0,r;Ω,
which is independent of ζ,qand ξ. Furthermore, it is easy to see that the null space associated with
the bilinear form bis given by (see, e.g. [21, eq. (3.35)] for the case pρ, ϱq“p4,4{3q)
Vb:“"pϑ, r
sq P H:żΩr
τ¨r
s`żΩ
ϑdivpr
τq “ 0@r
τPQ*
“!pϑ, r
sq P H:r
s“∇ϑand ϑPH1
0pΩq).
Then, following the same ideas as in [21, Lemma 3.6], we have to prove that a
q,ξ is Vb-elliptic plus an
inf-sup condition on b. To show the property of a
q,ξ, we use the above characterization along with
(2.10) and the continuous injection iρ: H1pΩq Ñ LρpΩq. In this way, for each 
ϑ“ pϑ, r
sq P Vb, we get
a
q,ξp
ϑ, 
ϑq ě }ϑ}2
0;Ω `κ0}r
s}2
0;Ω `żΩ
z¨r
sϑěr
κ0}ϑ}2
1;Ω ` pκ0{2q}r
s}2
0;Ω `żΩ
z¨r
sϑ
ěrκ0}iρ}´2}ϑ}2
0,ρ;Ω ` pκ0{2q }r
s}2
0;Ω ´ }z}0,r;Ω }r
s}0;Ω}ϑ}0,ρ;Ω
ě1
2p2κ´ }z}0,r;Ωq }
ϑ}2,
(4.12)
where the constants r
κ0and κare given by
rκ0:“min !κ0
2,1)and κ:“min !rκ0}iρ}´2,κ0
2).
16

Thus, under the assumption }z}X2ďαA:“2
3κ, the inequality (4.12) implies
a
q,ξp
ϑ, 
ϑq ě αA}
ϑ}2@
ϑ:“ pϑ, r
sq P Vb,(4.13)
which establishes the Vb-ellipticity of a
q,ξ with constant αA.
The inf-sup condition for the bilinear form bstates that there exists a constant r
βą0 such that
sup

ϑPH

ϑ‰0
bp
ϑ, r
τq
}
ϑ}H
ěr
β}r
τ}divϱ;Ω @r
τPQ,(4.14)
which can be proved analogously to the case pρ, ϱq“p4,4{3qprovided in [21, Lemma 3.3, eq. (3.45)]
since the present indexes ρand ϱare conjugate to each other as well.
The previous discussion allows us to establish the following lemma on the existence and uniqueness
of solution of the decoupled system (4.11).
Lemma 4.5. Let ρand ϱbe within the range of values stipulated by (3.27). Then, for each pζ,
z, ξq “
`ζ,pz, qq, ξ˘PX2ˆpX2ˆM1qˆH1such that }z} ď αA, there exists a unique p
θ, r
σq “ `pθ, r
tq,r
σ˘PHˆQ
solution of (4.11), and hence one can define Πpζ,
z, ξq:“
θ. Moreover, there exist positive constants
CΠand ¯
CΠ, depending on αA,r
β,|Ω|,ρ, and κ2, such that the following a priori estimates hold
}Πpζ,
z, ξq} “ }
θ}HďCΠ}g}0,ϱ;Ω ,}r
σ}Qď¯
CΠ}g}0,ϱ;Ω .(4.15)
Proof. The proof is a consequence of the Vb-ellipticity of a
q,ξ (cf. (4.13)), the inf-sup condition (4.14),
and a direct application of [6, Theorem 2.1, Corollary 2.1]. Note that the dependence of the constants
CΠand ¯
CΠon |Ω|,ρ, and κ2, is due to }a}(cf. (3.31)) since }a
q,ξ}, which is required by the abstract
a priori estimates from [6, Corollary 2.1, eqs. (2.15) and (2.16)], is bounded above by }a}`}z}.
4.4 Solvability of the fully-mixed formulation
In order to solve the fully-mixed coupled problem (3.34) we propose a fixed-point strategy based
on the operators S,Ξand Π, which correspond to the decoupled problems (4.1), (4.3) and (4.11),
respectively. The coupling of the three problems can be analyzed in terms of the compose operator
T:X2ˆH1ÑX2ˆH1given by
Tpζ, ϑq:“´S`Ξ2pζ, ϑq, ϑ˘,Π1`SpΞ2pζ, ϑq, ϑq,Ξpζ, ϑq, ϑ˘¯@ pζ, ϑq P X2ˆH1.(4.16)
The well-definedness of S,Ξand Π, which was obtained from Lemmas 4.1,4.4 and 4.5, respectively,
implies the same property for the operator T. Furthermore, due to the nonlinear character of Π, the
operator Tbecomes nonlinear as well. Then, we observe that solving (3.34) is equivalent to seeking a
fixed-point of T, that is: Find pρ, θq P X2ˆH1such that
Tpρ, θq “ pρ, θq.(4.17)
In what follows, we address the solvability of the nonlinear equation (4.17), equivalently of (3.34),
by means of the Banach fixed-point theorem. For this purpose, given δą0, we first introduce the ball
Wpδq:“!pζ, ϑq P X2ˆH1:}pζ, ϑq} :“ }ζ}X2` }ϑ}0,ρ;Ω ďδ).
Now, given pζ, ϑq P Wpδq, the definition of Tyields
}Tpζ, ϑq} “ ››S`Ξ2pζ, ϑq, ϑ˘››X2`››Π1`SpΞ2pζ, ϑq, ϑq,Ξpζ, ϑq, ϑ˘››0,ρ;Ω ,
17

from which, assuming (4.7) and the upper bound
}Ξ1pζ, ϑq}X2ďαA,(4.18)
and bearing in mind the a priori estimates for S,Ξand Π(cf. (4.2), (4.10) and (4.15), respectively),
we find that
}Tpζ, ϑq} ď CT!}uD}1{s,r;Γ ` }f}0,r;Ω ` }pD}1{s,r;Ω
` }f}0,r;Ω ` }g}0,ϱ;Ω `γpλq`}ζ}X2` }ϑ}0,ρ;Ω˘),
(4.19)
where CTis a positive constant depending on CS,CΞand CΠ. In turn, we deduce from the estimate
for }Ξpζ, ϑq} (cf. (4.10)) that a sufficient condition for the assumption (4.18) is given by
CΞ!}pD}1{s,r;Γ ` }f}0,r;Ω ` }uD}1{s,r;Γ `γpλq`}ζ}X2` }ϑ}0,ρ;Ω ˘)ďαA.
In this way, noting that certainly }ζ}X2` }ϑ}0,ρ;Ω ďδwe conclude the following result.
Lemma 4.6. Let ρ,ϱ,rand sbe the real numbers within the range specified in (3.27), and λąM.
Assume that the data are sufficiently small so that (4.7)and the conditions
CΞ!}pD}1{s,r;Γ ` }f}0,r;Ω ` }uD}1{s,r;Γ `γpλqδ)ďαA,and (4.20a)
CT!}uD}1{s,r;Γ ` }f}0,r;Ω ` }pD}1{s,r;Ω ` }f}0,r ;Ω ` }g}0,ϱ;Ω `γpλqδ)ďδ , (4.20b)
are satisfied. Then, the operator Tmaps the ball Wpδqinto itself, that is TpWpδqq Ď Wpδq.
We now aim to prove that the operator Tis Lipschitz-continuous, for which, according to its
definition (cf. (4.16)), it suffices to show that S,Ξand Πsatisfy the same property. We begin with
the corresponding result for S.
Lemma 4.7. Let rand sbe within the range of values stipulated by (3.27), and λąM. Then, with
the same constant CSfrom the a priori estimate (4.2)for S(cf. Lemma 4.1), there holds
}Spq1, ϑ1q ´ Spq2, ϑ2q}X2ďCSγpλq }pq1, ϑ1q´pq2, ϑ2q}M1ˆH1,(4.21)
for all pq1, ϑ1q,pq2, ϑ2q P M1ˆH1.
Proof. Given pq1, ϑ1q,pq2, ϑ2q P M1ˆH1, we let Spq1, ϑ1q “ ρ1PX2and Spq2, ϑ2q “ ρ2PX2, where
pρ1,u1qand pρ2,u2qin X2ˆM1are the respective unique solutions of (4.1). Then, thanks to the
linearity of this problem, it is straightforward to see that pρ1´ρ2,u1´u2q P X2ˆM1is the unique
solution of (4.1) with Fq1,ϑ1´Fq2,ϑ2and the null functional instead of Fq,ϑ and G, respectively.
Consequently, noting from (3.10a) that
`Fq1,ϑ1´Fq2,ϑ2˘pτq “ ´γpλqżΩ`αpq1´q2q ` βpϑ1´ϑ2q˘trpτq @ τPX1,
the a priori estimate (4.2) yields
}Spq1, ϑ1q ´ Spq2, ϑ2q}X2“ }ρ1´ρ2}X2ďCSγpλq`}q1´q2}0,r;Ω ` }ϑ1´ϑ2}0,ρ;Ω˘,
which ends the proof.
The Lipschitz continuity of the operator Ξis addressed next.
18

Lemma 4.8. Let rand sbe within the range of values stipulated by (3.27), and assume that the data
fulfills condition (4.7). Then, with the same constant CΞfrom the a priori estimate (4.10)for Ξ(cf.
Lemma 4.4), there holds
}Ξpζ1, ϑ1q ´ Ξpζ2, ϑ2q}X2ˆM1ďCΞγpλq }pζ1, ϑ1q´pζ2, ϑ2q}X2ˆH1,(4.22)
for all pζ1, ϑ1q,pζ2, ϑ2q P X2ˆH1.
Proof. The proof follows in a similar fashion as the previous lemma. Given the two pairs of functions
pζ1, ϑ1q,pζ2, ϑ2q P X2ˆH1, we let Ξpζ1, ϑ1q“pw1, p1q P X2ˆM1and Ξpζ2, ϑ2q “ pw2, p2q P X2ˆM1
in X2ˆM1, where pw1, p1qand pw2, p2qare the respective solutions of (4.3). Then, thanks to the
linearity of (4.3), we realize that pw1´w2, p1´p2q P X2ˆM1is the unique solution of this problem
when Gζ,ϑ and Fare replaced by Gζ1,ϑ1´Gζ2,ϑ2and the null functional, respectively. In this way,
noting from (3.21b) that
`Gζ1,ϑ1´Gζ2,ϑ2˘pqq “ c2pλqżΩ
trpζ1´ζ2qq`c3pλqżΩ
pϑ1,0´ϑ2,0qq
where ϑi,0“mpϑiq(cf. (3.13)), iP t1,2u, the a priori estimate (4.10) gives
}Ξpζ1, ϑ1q ´ Ξpζ2, ϑ2q} “ }w1´w2}X2` }p1´p2}M1
ďCΞγpλq´}ζ1´ζ2}X2` }ϑ1´ϑ2}0,ρ;Ω¯,
which concludes the proof.
It remains to establish the continuity of Π, for which, following the approach from several previous
works (see, e.g [26,28,29]), we assume from now on a regularity assumption on the solution of the
problem defining this operator, namely
pH.1qthere exists εěn
rand a positive constant r
Cε, such that for each pζ,
z, ξq P X2ˆpX2ˆM1qˆH1,
there hold Πpζ,
z, ξq:“
θ“ pθ, r
tq P Wε,ρpΩq ˆ HεpΩq, and
}
θ}:“ }θ}ε,ρ;Ω ` }r
t}ε;Ω ďr
Cε}g}0,ϱ;Ω .(4.23)
The aforementioned lower bound of εis explained within the proof of Lemma 4.9 below, which
provides the Lipschitz-continuity of Π. In this regard, we recall here that for each εăn
2there holds
HεpΩq Ă Lε˚pΩqwith continuous injection
iε:HεpΩq Ñ Lε˚pΩq,where ε˚“2n
n´2ε.
Note that the indicated lower and upper bounds for the additional regularity ε, which turn out to
require that εP rn
r,n
2q, are compatible if and only if rą2, which is coherent with the range stipulated
in (3.27). Thus, we have the following result.
Lemma 4.9. Let ρand ϱbe within the range of values stipulated by (3.27), and assume that the
regularity condition pH.1q pcf. (4.23)qholds. Then, there exists a positive constant LΠ, depending on
LK,αA,iε,r
Cε,|Ω|,r,n, and ε, such that
}Πpζ1,
z1, ξ1q ´ Πpζ2,
z2, ξ2q}HďLΠ}g}0,ϱ;Ω }pζ1,
z1, ξ1q´pζ2,
z2, ξ2q} ,(4.24)
for all pζ1,
z1, ξ1q “ `ζ1,pz1, q1q, ξ1˘,pζ2,
z2, ξ2q “ `ζ2,pz2, q2q, ξ2˘PX2ˆ pX2ˆM1q ˆ H1, such that
}z1}X2,}z2}X2ďαA.
19

Proof. Given pζ1,
z1, ξ1q,pζ2,
z2, ξ2q P X2ˆpX2ˆM1qˆ H1as indicated, we let 
θ1:“Πpρ1,
z1, ξ1q P H
and 
θ2:“Πpρ2,
z2, ξ2q P H, where p
θ1,r
σ1q “ `pθ1,r
t1q,r
σ1˘PHˆQand p
θ2,r
σ2q “ `pθ2,r
t2q,r
σ2˘P
HˆQare the respective solutions of (4.11). Defining 
q1:“ pζ1,
z1, q1qand 
q2:“ pζ2,
z2, q2q, it follows
from the corresponding second equation of (4.11) that 
θ1´
θ2PVb, and then the Vb-ellipticity of a
q1,ξ1
(cf. (4.13)) gives
}
θ1´
θ2}2
Hď1
αA
a
q1,ξ1p
θ1´
θ2,
θ1´
θ2q.(4.25)
In turn, the evaluation at 
θ1´
θ2of the two systems arising from (4.11) for the pairs p
q1, ξ1qand
p
q2, ξ2q, lead to
a
q1,ξ1p
θ1,
θ1´
θ2q “ Fp
θ1´
θ2qand a
q2,ξ2p
θ2,
θ1´
θ2q “ Fp
θ1´
θ2q,
from which we find that
a
q1,ξ1p
θ1´
θ2,
θ1´
ϑ2q “ a
q1,ξ1p
θ1,
θ1´
θ2q ´ a
q1,ξ1p
θ2,
θ1´
θ2q
“a
q2,ξ2p
θ2,
θ1´
θ2q ´ a
q1,ξ1p
θ2,
θ1´
θ2q
“żΩ
pKpζ2, q2, ξ2q ´ Kpζ1, q1, ξ1qqr
t2¨ pr
t1´r
t2q ` żΩ
pz2´z1q ¨ r
t2pθ1´θ2q.
(4.26)
Next, invoking the Lipschitz-continuity of K(cf. (2.11)), and making use of the Cauchy–Schwarz and
H¨older inequalities, we obtain
żΩ
pKpζ2, q2, ξ2q ´ Kpζ1, q1, ξ1qqr
t2¨ pr
t1´r
t2q
ďLK´}ζ2´ζ1}0,2t1;Ω ` }q1´q2}0,2t1;Ω ` }θ1´θ2}0,2t1;Ω¯}r
t2}0,2t;Ω }r
t1´r
t2}0;Ω ,
(4.27)
where t, t1P p1,`8q are conjugate to each other. Now, choosing tsuch that 2t“ε˚, we get 2t1“n
ε,
which, according to the range stipulated for ε, yields 2t1ďr, and certainly rďρ, so that the
norm of the embedding of the respective Lebesgue spaces is given by Cr,ε :“ |Ω|rε´n
rn . In this way,
using additionally the continuity of iεalong with the regularity assumption (4.23), the estimate (4.27)
becomes
żΩ
pKpζ2, q2, ξ2q ´ Kpζ1, q1, ξ1qqr
t2¨ pr
t1´r
t2q
ďr
LΠ}g}0,ϱ;Ω !}ζ1´ζ2}X2` }q1´q2}M1` }ξ1´ξ2}0,ρ;Ω)}
θ1´
θ2}H,
(4.28)
where r
LΠdepends on LK,Cr,ε,r
Cε,}iε}and |Ω|. In turn, bearing in mind the a priori estimation of
r
t2(cf. (4.15)), the Cauchy–Schwarz and H¨older inequalities yield
żΩ
pz2´z1q ¨ r
t2pϑ1´ϑ2q ď CΠ}g}0,ϱ;Ω }z2´z1}X2}
θ1´
θ2}H.(4.29)
Finally, replacing (4.28) and (4.29) back into (4.26), we deduce, along with (4.25), the required in-
equality (4.24) with LΠ:“1
αAmax␣r
LΠ, CΠ(, which ends the proof.
Now, we conclude that, under the hypotheses of Lemmas 4.7,4.8 and 4.9, the compose operator
T(cf. (4.16)) becomes Lipschitz-continuous within the ball Wpδqof the space X2ˆLρpΩq. This is
summarized in the next lemma.
20

Lemma 4.10. Let ρ,ϱ,rand sbe the real numbers within the range specified in (3.27), and λąM.
In addition, assume that the regularity condition pH.1q(cf. (4.23)) holds, and that the data are
sufficiently small so that (4.7),(4.20a), and (4.20b)are satisfied, that is
}e}:“max ␣χ , n α2γpλq(ďαA
2,
CΞ!}pD}1{s,r;Γ ` }f}0,r;Ω ` }uD}1{s,r;Γ `γpλqδ)ďαA,and
CT!}uD}1{s,r;Γ ` }f}0,r;Ω ` }pD}1{s,r;Ω ` }f}0,r ;Ω ` }g}0,ϱ;Ω `γpλqδ)ďδ .
Then, there exists a positive constant LT, depending on CS,CΞ, and LΠ, such that
}Tpζ1, ϑ1q ´ Tpζ2, ϑ2q}
ďLT´γpλq`γpλq`}g}0,ϱ;Ω ˘` }g}0,ϱ;Ω¯}pζ1, ϑ1q´pζ2, ϑ2q}X2ˆLρpΩq,(4.30)
for all pζ1, ϑ1q,pζ2, ϑ2q P Wpδq.
Proof. It readily follows from the definition of the operator T(cf. (4.16)), and the estimates (4.21),
(4.22), and (4.24).
We are now in position to formulate the main result of this section, which establishes the existence
of a unique fixed-point of T(cf. (4.17)), equivalently, the existence and uniqueness of solution of the
coupled system (3.34).
Theorem 4.11. Let ρ,ϱ,rand sbe the real numbers within the range specified in (3.27), and
λąM. In addition, assume that the regularity condition (H.1) (cf. (4.23)) holds, and that the data
are sufficiently small so that (4.7),(4.20a), and (4.20b)are satisfied. Besides, suppose that
LT´γpλq`γpλq`}g}0,ϱ;Ω ˘` }g}0,ϱ;Ω¯ă1,(4.31)
where LTis the positive constant from Lemma 4.10. Then, the operator Thas a unique fixed-point
pρ, θq P Wpδq. Equivalently, the coupled problem (3.34)has a unique solution pρ,uq P X2ˆM1,
pw, pq P X2ˆM1and p
θ, r
σq P HˆQ, with pρ, θq P Wpδq. Moreover, there hold
}pρ,uq}X2ˆM1ďr
CS!}uD}1{s,r;Γ ` }f}0,r;Ω ` }pD}1{s,r;Ω ` }f}0,r ;Ω ` }g}0,ϱ;Ω `γpλqδ),
}pw, pq}X2ˆM1ďr
CΞ!}uD}1{s,r;Γ ` }pD}1{2,Γ` }f}0,r;Ω ` }g}0,ϱ;Ω `γpλqδ),
}p
θ, r
σq}HˆQďr
CΠp1`δq }g}0,ϱ;Ω ,
where r
CS,r
CΞ, and r
CΠare positive constants depending on CS,CΞand CΠ.
Proof. Recall, from Lemma 4.6, that (4.20a) and (4.20b) guarantee that Tmaps Wpδqinto itself.
Hence, in virtue of the equivalence between (3.34) and (4.17), and bearing in mind the Lipschitz-
continuity of (4.30) (cf. Lemma 4.10) and the hypothesis (4.31), a straightforward application of the
Banach fixed point Theorem implies the existence of a unique solution pρ, θq P Wpδqof (3.34), and
hence, the existence of a unique pρ,uq P X2ˆM1,pw, pq P X2ˆM1and p
θ, r
σq P HˆQsolution of
(3.34). In addition, the a priori estimates follow straightforwardly from (4.2), (4.10) and (4.15), and
bounding }ρ}X2and }θ}X2by δ.
21

We would like to end this section by emphasizing that the hypothesis λąM(as used in Sections
4.1 and 4.4) naturally hold true in the context of the nearly incompressible scenario. Consequently,
we proceed by assuming that λis sufficiently large, which, in turn, makes γpλqto become sufficiently
small (cf. (2.4)). In this way, considering diminutive values for χ, we ensure the feasibility of (4.7).
A similar remark arises later on in the discrete analysis.
5 The discrete analysis
5.1 Preliminaries
Let tThuhą0be a regular family of triangulations Thof the domain Ω made of triangles Kin 2D (resp.
tetrahedra Kin 3D) with corresponding diameter hKą0. The meshsize h, which also stands for the
sub-index, is defined by the largest diameter of the triangulation Th, that is h:“max ␣hK:KPTh(.
Furthermore, we let PℓpSq(resp. ¯
PℓpSq) be the space of polynomials defined on SĂΩ of degree
ďℓPN(resp. “ℓ). The vector counterpart of PℓpSqis denoted by PℓpSq:“ rPℓpSqsn. In turn, for a
generic vector xPRn, we define the local Raviart–Thomas finite element space of order ℓover KPTh
as RTℓpKq:“PℓpKq ‘ ¯
PℓpKqx. Then, based on the above, we introduce the following global spaces
PℓpΩq:“!whPL2pΩq:wh|KPPℓpKq,@KPTh),
PℓpΩq:“!whPL2pΩq:wh|KPPℓpKq,@KPTh),
RTℓpΩq:“!τhPHpdiv; Ωq:τh|KPRTℓpKq,@KPTh),
RTℓpΩq:“!τhPHpdiv; Ωq:τh,i|KPRTℓpKq,@iP t1, ..., nu,@KPTh),
where τh,i stands for the ith-row of a tensor τh. It is easy to see that for each tP r1,`8s there hold
PℓpΩq Ď LtpΩq,PℓpΩq Ď LtpΩq,RTℓpΩq Ď Hpdivt; Ωq X Htpdivt; Ωq,
and RTℓpΩq Ď Hpdivt; Ωq X Htpdivt; Ωq.
5.2 The discrete coupled system
In order to set the discrete version of (3.34), we now resort to the definitions from Section 5.1 to
introduce the following sets of finite element subspaces, one for each decoupled problem:
X2,h :“Hr
0pdivr; Ωq X RTℓpΩq,X1,h :“Hs
0pdivs; Ωq X RTℓpΩq,M1,h :“PℓpΩq “:M2,h,(5.1a)
X2,h :“RTℓpΩq,X1,h :“RTℓpΩq,M1,h :“PℓpΩq “: M2,h,(5.1b)
H1,h :“PℓpΩq,H2,h :“PℓpΩq,Hh:“H1,h ˆH2,h ,Qh:“RTℓpΩq.(5.1c)
Then, the Galerkin scheme associated with (3.34) reads: Find pρh,uhq P X2,h ˆM1,h ,pwh, phq P
X2,h ˆM1,h and p
θh,r
σhq:“`pθh,r
thq,r
σh˘PHhˆQhsuch that
apρh,τhq ` b1pτh,uhq “ Fph,θhpτhq @ τhPX1,h ,
b2pρh,vhq “ Gpvhq @ vhPM2,h ,
cpwh,zhq ` d1pzh, phq “ Fpzhq @ zhPX1,h ,
d2pwh, qhq ´ epph, qhq “ Gρh,θhpqhq @ qhPM2,h ,
a
ph,θhp
θh,
ϑhq ` bp
ϑh,r
σhq “ Fp
ϑhq @ 
ϑhPHh,
bp
θh,r
τhq “ 0@r
τhPQh,
(5.2)
22

where 
ph:“ pρh,wh, phq P X2,h ˆX2,h ˆM1,h.
For the solvability analysis of (5.2) we will adopt a discrete version of the fixed-point strategy
developed in Section 4.4. To this end, we first use the analogues of the operators S,Ξ, and Πto
introduce in the following section the corresponding discrete decoupled problems, and establish their
well-posedness.
5.3 The discrete decoupled problems
We begin by letting Sh: M1,h ˆH1,h ÑXhbe the operator defined by
Shpqh, ϑhq:“ρh@ pqh, ϑhq P M1,h ˆH1,h ,
where pρh,uhq P X2,h ˆM1,h is the unique solution (to be confirmed below) of the discrete formulation
arising from the first and second rows of (5.2) after replacing pph, θhqby pqh, ϑhq, that is
apρh,τhq ` b1pτh,uhq “ Fqh,ϑhpτhq @ τhPX1,h ,
b2pρh,vhq “ Gpvhq @ vhPM2,h .(5.3)
For the solvability analysis of (5.3), we first observe from (5.1a) that
divpXi,hq Ď Hi,h @iP t1,2u,
whence the discrete kernels of b1and b2coincide, and are given by
Kℓ
h:“!τhPRTℓpΩq:divpτhq “ 0and żΩ
trpτhq “ 0).
Furthermore, since the bilinear forms involved in the mixed formulation of the poroelasticity equations
coincide with those of [28, eq. (3.15)], and additionally the same finite element subspaces (cf. (5.1a))
are employed here, in what follows we proceed to simply use the results from [28]. In this way, given
tP p1,`8q, we consider the mesh size hℓ
tfor which the usual L2pΩq-orthogonal projector satisfies the
property stated in [28, eq. (5.21)]. Then, thanks to [28, Lemma 5.3], there exist positive constants
Mdand αA,dsuch that for each λąMdand for each hďh0:“minthℓ
r, hℓ
suthere hold
sup
τhPKℓ
h
τh‰0
apζh,τhq
}τh}X1
ěαA,d}ζh}X2@ζhPX2,h ,
sup
ζhPKℓ
h
apζh,τhq ą 0@τhPKℓ
h,τh‰0.
(5.4)
In addition, the inf-sup conditions for the bilinear forms b1and b2, proved in [28, Lemma 5.4], provide
the existence of positive constants β1,dand β2,d, independent of h, such that
sup
τhPXi,h
τh‰0
bipτh,vhq
}τh}Xi
ěβi,d}vh}Mi@vPMi,h ,@iP t1,2u.(5.5)
Thus, thanks to (5.4) and (5.5), we are in position to show next the discrete version of Lemma 4.1.
Lemma 5.1. Let rand sbe within the range of values stipulated by (3.27), and λąMd. Then, for
each pqh, ϑhq P M1,h ˆH1,h there exists a unique pρh,uhq P X2,h ˆM1,h solution of (5.3), and hence
one can define Shpqh, ϑhq:“ρh. Moreover, there exists a positive constant CS,d, depending on αA,d,
β1,d,β2,d,CF, and µ, and hence independent of h, such that for each hďh0:“minthℓ
r, hℓ
suthere
holds
}Shpqh, ϑhq} “ }ρh}X2ďCS,d"}uD}1{s,r;Γ ` }f}0,r;Ω `γpλq´}qh}0,r;Ω ` }ϑh}0,ρ;Ω ¯*.(5.6)
23

Proof. It follows from a direct application of the discrete Babuˇska–Brezzi theory in Banach spaces (cf.
[6, Theorem 2.1, Corollary 2.1,]). Note that the dependence of the constant CS,don µis due to }a}
(cf. (3.11)).
We now let Ξh:X2,h ˆH1,h ÑM1,h be the operator defined by
Ξhpζh, ϑhq “ `Ξ1,hpζh, ϑhq,Ξ2,h pζh, ϑhq˘:“ pwh, phq @ pζh, ϑhq P X2,h ˆH1,h ,
where pwh, phq P X2,h ˆM1,h is the unique solution (to be confirmed below) of the discrete formulation
arising from the third and fourth rows of (5.2) after replacing pρh, θhqby pζh, ϑhq, that is
cpwh,zhq ` d1pzh, phq “ Fpzhq @ zhPX1,h ,
d2pwh, qhq ´epph, qhq “ Gζh,ϑhpqhq @ qhPM2,h .
(5.7)
Then, similarly as for (5.3), we first notice that
divpXi,hq Ď Mi,h @iP t1,2u,
which yields the discrete kernels of d1and d2to become
Vℓ
h:“!zhPRTℓpΩq: divpzhq “ 0).
Knowing the above, the discrete version of Lemma 4.2 is now recalled from [20, Lemma 5.2].
Lemma 5.2. Assume that rand ssatisfy the particular range specified by (3.27). Then, there exists
a positive constant αc,dsuch that
sup
zhPVℓ
h
zh‰0
cpwh,zhq
}zh}X1
ěαc,d}wh}X2@whPVℓ
h,
and
sup
wPVℓ
h
cpwh,zhq ą 0@zhPX1,h ,zh‰0.
Proof. It proceeds analogously to the proof of [29, Lemma 4.3]. However, for full details we refer to
[19, Lemma 5.2], which is the preprint version of [20].
On the other hand, the discrete inf-sup conditions for the bilinear forms d1and d2, which can be
found in [29, Lemma 5.3], state that for each iP t1,2u, there exists a positive constant r
βi,dsuch that
sup
zhPXi,h
zh‰0
dipzh, qhq
}zh}Xi,h
ěr
βi,d}qh}Mi@qhPMi,h .(5.8)
Then, analogously to the continuous case, Lemma 5.2 and (5.8) imply that the bilinear form r
A(cf.
(4.4)) satisfies the global inf-sup conditions given by the discrete versions of (4.5a) and (4.5b), both
with a positive constant αA,ddepending on αc,d,r
β1,d,r
β2,d, and }c}, and hence independent of h.
Moreover, using these inequalities, and proceeding analogously to the derivation of (4.8) and (4.9),
which means assuming now the discrete version of (4.7), this is
}e} “ max ␣χ, n α2γpλq(ďαA,d
2,(5.9)
24

we arrive at the discrete global inf-sup conditions for the global operator A(cf. (4.6)), namely
sup
pzh,qhqPX1,hˆM2,h
pzh,qhq‰0
Appwh, phq,pzh, qhqq
}pzh, qhq}X1ˆM2
ěαA,d
2}pwh, phq}X2ˆM1@ pwh, phq P X2,h ˆM1,h ,
(5.10a)
sup
pwh,phqPX2,hˆM1,h
pwh,phq‰0
Appwh, phq,pzh, qhqq
}pwh, phq}X2ˆM1
ěαA,d
2}pzh, qhq}X1,hˆM2,h @ pzh, qhq P X1,h ˆM2,h .
(5.10b)
Similarly as for the continuous analysis, we stress here that the fact that γpλqapproaches 0 as λ
increases (cf. (2.4)), ensures the feasibility of (5.9) for sufficiently large λand sufficiently small χ.
Having established (5.10a) and (5.10b), a straightforward application of the discrete version of the
Banach–Neˇcas–Babuˇska theorem (cf. [24, Theorem 2.22]) allows to conclude the following result.
Lemma 5.3. Let rand sbe within the range of values specified by (3.27), and assume that the data
satisfy (5.9). Then, for each pζh, ϑhq P X2,h ˆH1,h there exists a unique pwh, phq P X2,h ˆM1,h
solution of (5.7), and hence one can define Ξhpζh, ϑhq “ `Ξ1,hpζh, ϑhq,Ξ2,hpζh, ϑhq˘:“ pwh, phq P
X2,h ˆM1,h. Moreover, there exists a positive constant CΞ,d, depending on αA,d,CF, and CG, and
hence independent of h, such that
}Ξhpζh, ϑhq}X2ˆM1“ }wh}X2` }ph}M1
ďCΞ,d"}pD}1{s,r;Γ ` }f}0,r;Ω ` }uD}1{s,r;Γ `γpλq´}ζh}X2` }ϑh}0,ρ;Ω ¯*.(5.11)
Finally, we let Πh:X2,h ˆ pX2,h ˆM1,hq ˆ H1,h ÑHhbe the operator defined by
Πhpζh,
zh, ξhq “ `Π1,hpζh,
zh, ξhq,Π2,hpζh,
zh, ξhq˘:“
θh“ pθh,r
thq,
for all pζh,
zh, ξhq “ `ζh,pzh, qhq, ξh˘PX2,h ˆ pX2,h ˆM1,hq ˆ H1,h , where p
θh,r
σhq P HhˆQhis the
unique solution (to be confirmed below) of the discrete formulation arising from the fifth and sixth
rows of (5.2) after replacing a
ph,θh, with 
ph:“ pρh,ωh, phq, by a
qh,ξh, with 
qh:“ pζh,zh, qhq, that is
a
qh,ξhp
θh,
ϑhq ` bp
ϑh,r
σhq “ Fp
ϑhq @ 
ϑh:“ pϑh,r
shq P Hh,
bp
θh,r
τhq “ 0@r
τhPQh.(5.12)
For the analysis of the Galerkin scheme (5.12), we proceed as in [21, Section 5.5] (see also [5, Section
4.3, Lemma 4.2] or [17, Section 5.3, eqs. (5.19), (5.20)]). More precisely, since the required results
are already available in those references, in what follows we just describe the main aspects of the
corresponding discussion, for which we first introduce the discrete kernel Vℓ
b,h of b(cf. (3.30b)), that
is
Vℓ
b,h :“!
ϑh:“ pϑh,r
shq P Hh:bp
ϑh,r
τhq “ 0@r
τhPQh),
and the subspace of Qhgiven by
Zℓ
b,h :“!r
τhPQh: divpr
τhq “ 0 in Ω).
Then, applying the abstract result provided in [21, Lemma 5.1], one deduces that the existence of
positive constants β1,dand β2,d, independent of h, such that
sup
r
τhPQh
r
τh“0
żΩ
ϑhdivpr
τhq
}r
τh}divϱ;Ω
ěβ1,d}ϑh}0,ρ;Ω @ϑhPH1,h ,and (5.13)
25

sup
r
shPH2,h
r
sh“0
żΩr
sh¨r
τh
}r
sh}0,Ω
ěβ2,d}r
τh}divϱ;Ω @r
τhPZℓ
b,h ,(5.14)
is equivalent to the existence of positive constants r
βdand r
Cd, independent of h, such that
sup

ϑhPHh

ϑh“0
bp
ϑh,r
τhq
}
ϑh}H
ěr
βd}r
τh}divϱ;Ω @r
τhPQh,and (5.15a)
}r
sh}0,Ωěr
Cd}ϑh}0,ρ;Ω @
ϑh:“ pϑh,r
shq P Vℓ
b,h .(5.15b)
The proof of (5.13) is basically provided at the last part of [21, Section 5.5] by noticing that it reduces
to the vector version of [21, Lemma 5.5]. Actually, while the proof there is for pρ, ϱq“p4,4{3q, it
can be extended almost verbatim to an arbitrary conjugate pair pρ, ϱqsatisfying (3.27). In turn, it
is readily seen that (5.14) follows from the fact that Zℓ
b,h ĎH2,h (cf. [17, eq. (5.18)]). In this way,
having already the discrete inf-sup condition (5.15a) for b, it only remains to employ (5.15b) to show
the Vℓ
b,h-ellipticity of a
qh,ξhfor given 
qh“ pζh,zh, qhq P X2,h ˆX2,h ˆM1,h and ξhPH1,h. Indeed,
proceeding similarly to the first part of the derivation of (4.12), we have for each 
ϑh:“ pϑh,r
shq P Vℓ
b,h
a
qh,ξhp
ϑh,
ϑhq ě pκ0{2qr
C2
d}ϑh}2
0,ρ;Ω ` pκ0{2q }r
sh}2
0;Ω ´ }zh}0,r;Ω }r
sh}0;Ω }ϑh}0,ρ;Ω
ě1
2!κ0min ␣1,r
C2
d(´ }zh}0,r;Ω)}
ϑh}2,
so that, under the constraint }zh}0,r;Ω ďαA,d:“1
3κ0min ␣1,r
C2
d(, there holds
a
qh,ξhp
ϑh,
ϑhq ě αA,d}
ϑh}2@
ϑh:“ pϑh,r
shq P Vℓ
b,h ,(5.16)
thus confirming the announced property of a
qh,ξh.
Hence, the solvability of (5.12) and therefore the well-posedness of Πhcan be established in the
following lemma.
Lemma 5.4. Let ρand ϱbe within the range of values stipulated by (3.27). Then, for each pζh,
zh, ξhq
“`ζh,pzh, qhq, ξh˘PX2,h ˆ pX2,h ˆM1,hq ˆ H1,h such that }zh} ď αA,d, there exists a unique
p
θh,r
σhq “ `pθh,r
thq,r
σh˘PHhˆQhsolution of (5.12), and hence one can define Πhpζh,
zh, ξhq “ 
θh.
Moreover, there exist positive constants CΠ,dand ¯
CΠ,d, depending on αA,d,r
βd,|Ω|,ρ, and κ2, and
hence independent of h, such that the following a priori estimates hold
}Πhpζh,
zh, ξhq} “ }
θh}HďCΠ,d}g}0,ϱ;Ω ,}r
σh}Qď¯
CΠ,d}g}0,ϱ;Ω .(5.17)
Proof. The result is a consequence of the Vℓ
b,h-ellipticity of a
qh,ξh(cf. (5.16)), the inf-sup condition
(5.15a), and a direct application of, for instance, [24, Theorem 2.34, Proposition 2.42]. Note that
the dependence of the constants CΠ,dand ¯
CΠ,don |Ω|,ρ, and κ2, is due to }a}(cf. (3.31)) since
}a
qh,ξh}, which is required by the theoretical estimates from [6, Corollary 2.2, eqs. (2.24) and (2.25)],
is bounded above by }a}`}zh}.
5.4 Solvability analysis of the discrete coupled system
The solvability analysis of the fully coupled discrete system (5.2) is performed in a similar fashion as
in the continuous case by using a fixed-point strategy, but now applying the Brouwer theorem instead
26

of the classical Banach one. Therefore, the structure and reasoning followed in this part, are going to
resemble partially the ones of Section 4.4. We begin this analysis by defining the discrete fixed-point
operator Th:X2,h ˆH1,h ÑX2,h ˆH1,h given by
Thpζh, ϑhq:“´Sh`Ξ2,hpζh, ϑhq, ϑh˘,Π1,h `ShpΞ2,hpζh, ϑhq, ϑhq,Ξhpζh, ϑhq, ϑh˘¯,(5.18)
for all pζh, ϑhq P X2,h ˆH1,h. Then, showing existence of solution is equivalent to seeking a fixed-point
to the operator Th, that is: Find pζh, ϑhq P X2,h ˆH1,h such that
Thpζh, ϑhq “ pζh, ϑhq.(5.19)
Now, given δą0, we define the δ-ball in the finite-dimensional subspace X2,h ˆH1,h by
Whpδq:“!pζh, ϑhq P X2,h ˆH1,h :}pζh, ϑhq} :“ }ζh}X2` }ϑh}0,ρ;Ω ďδ),
where we conveniently choose δ:“αA,d. Furthermore, assumption (5.9) applies to the discrete operator
Ξ1,h in the same way as (4.18) applies to Ξ1, this is
}Ξ1,hpζh, ϑhq}X2ďαA,d@ pζh, ϑhq P Whpδq.(5.20)
Combining the estimates (5.6), (5.11), and (5.17), we obtain the discrete version of (4.19) as an priori
bound for the operator Th, that is
}Thpζh, ϑhq} ď CT,d!}uD}1{s,r;Γ ` }f}0,r;Ω ` }pD}1{s,r;Ω
` }f}0,r;Ω ` }g}0,ϱ;Ω `γpλq`}ζh}X2` }ϑh}0,ρ;Ω˘),
where CT,dis a positive constant depending on CS,d,CΞ,dand CΠ,d, and hence independent of h.
In addition, taking into account the a priori estimate (5.11) with pζh, ϑhq P Whpδq, we conclude that
operator Ξ1,h will satisfy assumption (5.20) if there holds
CΞ,d!}pD}1{s,r;Γ ` }f}0,r;Ω ` }uD}1{s,r;Γ `γpλqδ)ďαA,d.
Hence, the following lemma establishes the conditions under which the operator Thmaps the ball
Whpδqinto itself, thus yielding the discrete analogue of Lemma 4.6.
Lemma 5.5. Let ρ,ϱ,rand sbe as specified in (3.27), and λąM. Moreover, assume that hď
h0:“minthℓ
r, hℓ
su, and that the data are sufficiently small so that (5.9)and the conditions
CΞ,d!}pD}1{s,r;Γ ` }f}0,r;Ω ` }uD}1{s,r;Γ `γpλqδ)ďαA,d,and (5.21a)
CT,d!}uD}1{s,r;Γ ` }f}0,r;Ω ` }pD}1{s,r;Ω ` }f}0,r ;Ω ` }g}0,ϱ;Ω `γpλqδ)ďδ , (5.21b)
are satisfied. Then, ThpWhpδqq Ď Whpδq.
The next two lemmas show, respectively, that the operators Shand Ξhare Lipschitz-continuous.
Lemma 5.6. Let rand sbe within the range of values stipulated by (3.27), and λąM. Then, with
the same constant CS,dfrom the a priori estimate (5.6)(cf. Lemma 5.1), for hďh0:“minthℓ
r, hℓ
su
there holds
}Shpq1,h, ϑ1,h q ´ Shpq2,h, ϑ2,h q}X2ďCS,dγpλq }pq1,h, ϑ1,hq´pq2,h, ϑ2,h q}M1ˆH1,(5.22)
for all pq1,h , ϑ1,hq,pq2,h , ϑ2,hq P M1ˆH1,h.
27

Proof. It proceeds analogously to the proof of Lemma 4.7. We omit further details.
Lemma 5.7. Let rand sbe within the range of values stipulated by (3.27), and assume that the
data fulfills condition (5.9). Then, with the same constant CΞ,dfrom the a priori estimate (5.11)(cf.
Lemma 5.3), there holds
}Ξhpζ1,h, ϑ1,h q ´ Ξhpζ2,h, ϑ2,h q}X2ˆM1ďCΞ,dγpλq }pζ1,h, ϑ1,hq´pζ2,h, ϑ2,h q}X2ˆH1,(5.23)
for all pζ1,h , ϑ1,hq,pζ2,h , ϑ2,hq P X2,h ˆH1,h.
Proof. It proceeds analogously to the proof of Lemma 4.8. Further details are omitted.
The next result shows the continuity of Πh. In this regard, we stress in advance that the obvious
absence of a regularity assumption in the present discrete setting, stops us of proving a Lipschitz-
continuity property of Πh.
Lemma 5.8. Let ρand ϱbe within the range of values stipulated by (3.27). Then, there exists a
positive constant LΠ,d, depending on LK,αA,d,|Ω|,r, and ρ, and hence independent of h, such that
}Πhpζ1,h,
ω1,h, ξ1,h q ´ Πhpζ2,h,
ω2,h, ξ2,h q}H
ďLΠ,d}Π2,hpζ2,h ,
ω2,h, ξ2,h q}0,ρ;Ω }pζ1,h,
ω1,h, ξ1,h q´pζ2,h ,
ω2,h, ξ2,h q} ,
(5.24)
for all pζ1,h ,
z1,h, ξ1,h q “ `ζ1,h,pz1,h , q1,hq, ξ1,h˘,pζ2,h,
z2,h, ξ2,h q “ `ζ2,h,pz2,h , q2,hq, ξ2,h˘PX2,h ˆ
pX2,h ˆM1,hq ˆ H1,h , such that }z1,h}X2,}z2,h }X2ďαA,d.
Proof. The proof follows similarly to the one of Lemma 4.9, except for the fact, as already announced,
that no regularity result can be applied. Indeed, given pζ1,h,
z1,h, ξ1,h qand pζ2,h,
z2,h, ξ2,h qas indicated,
we let 
ϑ1,h :“Πhpζ1,h,
z1,h, ξ1,h q P Hhand 
ϑ2,h :“Πhpζ2,h,
z2,h, ξ2,h q P Hh, where p
ϑ1,h,r
σ1,hq P Hhˆ
Qhand p
ϑ2,h,r
σ2,hq P HhˆQhare the respective solutions of (5.12). Defining 
q1,h :“ pζ1,h,
z1,h, q1,h q
and 
q2,h :“ pζ2,h,
z2,h, q2,h q, it follows from the corresponding second equation of (5.12) that 
ϑ1,h ´

ϑ2,h PVℓ
b,h, and then the Vℓ
b,h-ellipticity of a
q1,h,ξ1,h (cf. (5.16)) yields
}
ϑ1,h ´
ϑ2,h}2
Hď1
αA,d
a
q1,h,ξ1,h p
ϑ1,h ´
ϑ2,h,
ϑ1,h ´
ϑ2,hq.(5.25)
Then, proceeding analogously as for the derivation of (4.26), but now certainly employing p
q1,h, ξ1,h q,
p
q2,h, ξ2,h q, and (5.12), we obtain
a
q1,h,ξ1,h p
ϑ1,h ´
ϑ2,h,
ϑ1,h ´
ϑ2,hq “ żΩ
pz2,h ´z1,hq ¨ r
t2,hpϑ1,h ´ϑ2,h q
`żΩ
pKpζ2,h, q2,h , ξ2,hq ´ Kpζ1,h , q1,h, ξ1,hqq r
t2,h ¨ pr
t1,h ´r
t2,hq.
(5.26)
Next, using the Lipschitz-continuity of Kas in the estimate (3.2), recalling that r“2j,ρ“2k, and
rďρ, and noting that }r
t1,h ´r
t2,h}0;Ω ď }
ϑ1,h ´
ϑ2,h}H, we find that
żΩ
pKpζ2,h, q2,h , ξ2,hq ´ Kpζ1,h , q1,h, ξ1,hqq r
t2,h ¨ pr
t1,h ´r
t2,hq ď r
LK´}ζ1,h ´ζ2,h}0,r;Ω
` }q1,h ´q2,h}0,r;Ω ` }ξ1,h ´ξ2,h }0,ρ;Ω¯}r
t2,h}0,ρ;Ω }
ϑ1,h ´
ϑ2,h}H,
(5.27)
28

where r
LKdepends on LK,|Ω|,r, and ρ. In turn, the Cauchy–Schwarz and H¨older inequalities, and
the fact that
}ϑ1,h ´ϑ2,h}0,Ωď |Ω|pρ´2q{ρ}ϑ1,h ´ϑ2,h }0,ρ;Ω ď |Ω|pρ´2q{ρ}
ϑ1,h ´
ϑ2,h}H,
yield
żΩ
pz2,h ´z1,hq ¨ r
t2,h pϑ1,h ´ϑ2,hq ď |Ω|pρ´2q{ρ}z2,h ´z1,h }0,r;Ω }r
t2,h}0,ρ;Ω }
ϑ1,h ´
ϑ2,h}H.(5.28)
Finally, using (5.28) and (5.27) we can bound (5.26), so that the resulting estimate along with (5.25)
and the fact that Π2,hpζ2,h ,
z2,h, ξ2,h q “ r
t2,h, imply (5.24) and conclude the proof.
Combining Lemmas 5.6,5.7, and 5.9, we prove next that the operator This continuous in the
closed ball Whpδqof the space X2,h ˆH1,h. In order to simplify the corresponding statement and
proof, we let r
Shand r
Πh:“`r
Π1,h,r
Π2,h˘be the operators defined for each pζh, ϑhq P X2,h ˆH1,h by
r
Shpζh, ϑhq:“Sh`Ξ2,hpζh, ϑhq, ϑh˘and (5.29a)
r
Πhpζh, ϑhq:“Πh`r
Shpζh, ϑhq,Ξhpζh, ϑhq, ϑh˘,(5.29b)
so that r
Π1,h and r
Π2,h are obtained from (5.29b) by using, respectively, Π1,h and Π2,h instead of Πh.
Lemma 5.9. Let ρ,ϱ,rand sbe the real numbers within the range specified in (3.27), and λąM.
Moreover, assume that hďh0:“minthℓ
r, hℓ
su, and that the data are sufficiently small so that there
hold (5.9),(5.21a)and (5.21b). Then, there exists a positive constant LT,d, depending on CS,d,CΞ,d,
and γpλq, and hence independent of h, such that
}Thpζ1,h, ϑ1,h q ´ Thpζ2,h, ϑ2,h q}
ďLT,d`1`LΠ,d}r
Π2,hpζ2,h , ϑ2,hq}0,ρ;Ω ˘}pζ1,h, ϑ1,hq´pζ2,h, ϑ2,h q} ,
(5.30)
for all pζ1,h , ϑ1,hq,pζ2,h , ϑ2,hq P Whpδq.
Proof. Given pζ1,h, ϑ1,hq,pζ2,h , ϑ2,hq P Whpδq, we first observe from (5.18), (5.29a), and (5.29b) that
Thpζi,h, ϑi,h q “ `r
Shpζi,h, ϑi,h q,r
Π1,hpζi,h , ϑi,hq˘@iP␣1,2(,
which yields
}Thpζ1,h, ϑ1,h q ´ Thpζ2,h, ϑ2,h q}
ď }r
Shpζ1,h, ϑ1,h q ´ r
Shpζ2,h, ϑ2,h q} ` } r
Πhpζ1,h, ϑ1,h q˘´r
Πhpζ2,h, ϑ2,h q˘}.
(5.31)
Then, employing (5.29b) and (5.24), we find that
}r
Πhpζ1,h, ϑ1,h q˘´r
Πhpζ2,h, ϑ2,h q˘}
ďLΠ,d}r
Π2,hpζ2,h , ϑ2,hq}0,ρ;Ω !}r
Shpζ1,h, ϑ1,h q ´ r
Shpζ2,h, ϑ2,h q}
` }Ξhpζ1,h, ϑ1,h q ´ Ξhpζ2,h, ϑ2,h q} ` }ϑ1,h ´ϑ2,h}),
(5.32)
whereas (5.29a) and (5.22) imply
}r
Shpζ1,h, ϑ1,h q ´ r
Shpζ2,h, ϑ2,h q}
ďCS,dγpλq!}Ξhpζ1,h, ϑ1,h q ´ Ξhpζ2,h, ϑ2,h q} ` }ϑ1,h ´ϑ2,h}).
(5.33)
29

In this way, replacing (5.33) back into (5.32) and (5.31), and the resulting (5.32) back into (5.31) as
well, and performing minor algebraic manipulations, we arrive at
}Thpζ1,h, ϑ1,h q ´ Thpζ2,h, ϑ2,h q} ď `1`LΠ,d|r
Π2,hpζ2,h , ϑ2,hq}0,ρ;Ω ˘`1`CS,dγpλq˘
ˆ!}Ξhpζ1,h, ϑ1,h q ´ Ξhpζ2,h, ϑ2,h q} ` }ϑ1,h ´ϑ2,h}).
(5.34)
Finally, (5.34) and (5.23) give (5.30) with LT,d:“`1`CS,dγpλq˘`1`CΞ,dγpλq˘, and end the proof.
The main result of this section, which establishes the existence of solution of the discrete fixed-point
equation (5.19), or equivalently of the discrete coupled system (5.2), is presented now.
Theorem 5.10. Let ρ,ϱ,rand sbe the real numbers within the range specified in (3.27), and
λąM. Moreover, assume that hďh0:“minthℓ
r, hℓ
su, and that the data are sufficiently small so
that there hold (5.9),(5.21a)and (5.21b). Then, the operator Thhas a fixed-point pρh, θhq P Whpδq.
Equivalently, the coupled problem (5.2)has a solution pρh,uhq P X2,h ˆM1,h,pwh, phq P X2,h ˆM1,h ,
and p
θh,r
σhq P HhˆQh, with pρh, θhq P Whpδq. Moreover, there hold
}pρh,uhq}X2ˆM1ďr
CS,d!}uD}1{s,r;Γ ` }f}0,r;Ω ` }pD}1{s,r;Ω ` }f}0,r ;Ω ` }g}0,ϱ;Ω `γpλqδ),
}pwh, phq}X2ˆM1ďr
CΞ,d!}uD}1{s,r;Γ ` }pD}1{2,Γ` }f}0,r;Ω ` }g}0,ϱ;Ω `γpλqδ),
}p
θh,r
σhq}HˆQďr
CΠ,dp1`δq}g}0,ϱ;Ω ,
where r
CS,d,r
CΞ,dand r
CΠ,dare constants depending on CS,d,CΞ,dand CΠ,d.
Proof. From the assumptions (5.21a) and (5.21b), and Lemma 5.5 we have that Thmaps Whpδq
into itself. Furthermore, bearing in mind the continuity of Th(cf. Lemma 5.9), a straightforward
application of the Brouwer Theorem implies the existence of a solution pρh, θhq P Whpδqof (5.19), and
hence, the existence of pρh,uhq P X2,h ˆM1,h,pwh, phq P X2,h ˆM1,h and p
θh,r
σhq P HhˆQhsolution
of (5.2). Finally, the a priori estimates follows straightforwardly from (5.6), (5.11), and (5.17), and
bounding }ρh}X2and }θh}X2by δ.
5.5 A priori error analysis
The goal of this section is to establish an a priori error estimate for the Galerkin scheme (5.2). More
precisely, we are interested in deriving the usual C´ea estimate for the global error
E:“ }pρ,uq´pρh,uhq}X2ˆM1` }pω, pq´pωh, phq}X2ˆM1` }p
θ, r
σq´p
θh,r
σhq}HˆQ,
where `pρ,uq,pw, pq,p
θ, r
σq˘P pX2ˆM1qˆpX2ˆM1qˆpHˆQq, with pρ, θq P Wpδq, is the unique
solution of (3.34), which is guaranteed by Theorem 4.11, and `pρh,uhq,pwh, phq,p
θh,r
σhq˘P pX2,h ˆ
M1,hqˆpX2,h ˆM1,h qˆpHhˆQhq, with pρh, θhq P Whpδq, is a solution of (5.2), which is guaranteed
by Theorem 5.10. To this end, we proceed as in [20, Section 4.3] and apply suitable Strang estimates
to each one of the three pairs of associated continuous and discrete formulations forming (3.34) and
(5.2). Throughout the rest of this section, given a subspace Zhof a generic Banach space pZ, }¨}Zq,
we set for each zPZ
distpz, Zhq:“inf
zhPZh
}z´zh}Z.
We begin the analysis by applying the Strang estimate provided by [6, Proposition 2.1, Corollary
2.3, Theorem 2.3] to the context given by the first and second rows of (3.34) and (5.2). In this way,
30

we deduce the existence of a positive constant p
CS, depending on αA,d,β1,d,β2,d,}a},}b1}, and }b2}
(cf. (3.11), Section 5.3), such that there holds
}pρ,uq´pρh,uhq}X2ˆM1ďp
CS!dist`pρ,uq,X2,h ˆM1,h˘` }Fp,θ ´Fph,θh}X1
1,h ).(5.35)
Then, according to the definition of Fq,ϑ (cf. (3.10a)), we have that
`Fp,θ ´Fph,θh˘pτhq “ ´γpλqżΩ`αpp´phq ` βpθ´θhq˘trpτhq @ τhPX1,h ,
from which, applying H¨older’s inequality, and using that rďρ, we find that there exists a positive
constant s
CF, depending on n,r,ρ,|Ω|,α, and β, such that
}Fp,θ ´Fph,θh}X1
1,h ďs
CFγpλq!}p´ph}0,r;Ω ` }θ´θh}0,ρ;Ω).(5.36)
Next, we apply the classical first Strang’s Lemma (cf. [24, Lemma 2.27]) to the context given by
the third and fourth rows of (3.34) and (5.2). As a consequence, we obtain a positive constant p
CΞ,
depending on αA,d,}c},}d1},}d2}, and }e}(cf. (3.22), Section 5.3), such that there holds
}pω, pq´pωh, phq}X2ˆM1ďp
CΞ!dist`pω, pq,X2,h ˆM1,h˘` }Gρ,θ ´Gρh,θh}M1
2,h ).(5.37)
In this case, the definition of Gζ,ϑ (cf. (3.21b)) yields
`Gρ,θ ´Gρh,θh˘pqhq “ c2pλqżΩ
trpρ´ρhqqh`c3pλqżΩ
pθ0´θh,0qqh@qhPM2,h ,
so that, employing again H¨older’s inequality and the inequality rďρ, and bearing in mind the
definitions of the constants c2pλqand c3pλq(cf. (2.5)), we deduce that
}Gρ,θ ´Gρh,θh}M1
2,h ďs
CGγpλq!}ρ´ρh}0,r;Ω ` }θ´θh}0,ρ;Ω),(5.38)
where s
CGis a positive constant depending on n,r,ρ,|Ω|,α, and β.
Furthermore, we apply the Strang estimate provided by [21, Lemma 6.1] to the context given by
the fifth and sixth rows of (3.34) and (5.2). As a result, we get a positive constant p
CΠ, depending on
αA,d,r
βd,}ap,θ },}aph,θh}, and }b}(cf. (3.31), (3.32), Section 5.3), such that there holds
}p
θ, r
σq´p
θh,r
σhq}HˆQďp
CΠ!dist`p
θ, r
σq,HhˆQh˘` }a
p,θp
θ, ¨q ´ a
ph,θhp
θ, ¨q}H1
h),(5.39)
where 
p“ pρ,ω, pqand 
ph“ pρh,ωh, phq. Note that, being }ω}and }ωh}bounded by αAand αA,d,
it turns out that }ap,θ }and }aph,θh}are bounded by }a} ` αAand }a} ` αA,d, respectively. Now,
according to the definition of a
q,ξ (cf. (3.30a)), we have for all 
ϑh“ pϑh,r
shq
a
p,θp
θ, 
ϑhq ´ a
ph,θhp
θ, 
ϑhq “ żΩ!Kpρ, p, θq ´ Kpρh, ph, θhq)r
t¨r
sh`żΩ`ω´ωh˘¨r
tϑh.(5.40)
Regarding the first term on the right hand side of (5.40), we proceed exactly as for the derivation of
(4.28), so that, employing again the Lipschitz-continuity of K(cf. (2.11)), the Cauchy–Schwarz and
H¨older inequalities, the fact that rďρ, and the regularity assumption pH.1q(cf. (4.23)), we obtain
with the same constant r
LΠfrom (4.28) that
ˇˇˇˇżΩ!Kpρ, p, θq ´ Kpρh, ph, θhq)r
t¨r
shˇˇˇˇ
ďr
LΠ}g}0,ϱ;Ω !}ρ´ρh}0,r;Ω ` }p´ph}0,r;Ω ` }θ´θh}0,ρ;Ω )}r
sh}0,Ω.
(5.41)
31

In turn, proceeding similarly to the deduction of (4.29), which means using the above mentioned
classical inequalities, along with the a priori estimate (4.15), we can write with the same constant CΠ
from (4.15) that ˇˇˇˇżΩ`ω´ωh˘¨r
tϑhˇˇˇˇďCΠ}g}0,ϱ;Ω }ω´ωh}0,r;Ω }ϑh}0,ρ;Ω .(5.42)
Hence, utilizing the bounds provided by (5.42) and (5.41), we readily conclude from (5.40) that
}a
p,θp
θ, ¨q ´ a
ph,θhp
θ, ¨q}H1
h
ďs
Ca}g}0,ϱ;Ω !}ρ´ρh}0,r;Ω ` }ω´ωh}0,r;Ω ` }p´ph}0,r;Ω ` }θ´θh}0,ρ;Ω ),
(5.43)
where s
Ca:“max ␣r
LΠ, CΠ(. In this way, replacing (5.43) back into (5.39), (5.38) back into (5.37),
and (5.36) back into (5.35), and then adding the resulting inequalities, we arrive at
Eďp
C1!dist`pρ,uq,X2,h ˆM1,h˘`dist`pω, pq,X2,h ˆM1,h ˘
`dist`p
θ, r
σq,HhˆQh˘)`!p
C2γpλq ` p
C3}g}0,ϱ;Ω)E,
(5.44)
where p
C1:“max ␣p
CS,p
CΞ,p
CΠ(,p
C2:“max ␣p
CSs
CF,p
CΞs
CG(, and p
C3:“p
CΠs
Ca.
The announced C´ea estimate can be stated now.
Theorem 5.11. In addition to the hypotheses of Theorems 4.11 and 5.10, assume that
p
C2γpλq ` p
C3}g}0,ϱ;Ω ď1
2.(5.45)
Then, denoting p
C“2p
C1, there holds
}pρ,uq´pρh,uhq}X2ˆM1` }pω, pq´pωh, phq}X2ˆM1` }p
θ, r
σq´p
θh,r
σhq}HˆQ
ďp
C!dist`pρ,uq,X2,h ˆM1,h˘`dist`pω, pq,X2,h ˆM1,h ˘`dist`p
θ, r
σq,HhˆQh˘).
Proof. It readily follows after employing the assumption (5.45) in (5.44).
We now aim to establish the associated rates of convergence of the Galerkin scheme (5.2), for which
we collect approximation properties of the finite element subspaces that were introduced in Section
5.2. Indeed, thanks to the error estimates of the vector and tensor versions of the Raviart–Thomas
interpolator (see, e.g. [29, Section 4.1, eq. (4.6)]), as well as of the scalar and vector versions of the
L2-type projector onto piecewise polynomial spaces (see, e.g. [24, Proposition 1.135]), and due to
interpolation estimates of Sobolev spaces, there hold the following:
pAPρ
hqthere exists a positive constant C, independent of h, such that for each kP r1, ℓ `1s, and for
each τPWk,rpΩq X Hr
0pdivr; Ωq, with divpτq P Wk,r pΩq, there holds
distpτ,X2,hq:“inf
τhPX2,h
}τ´τh}r,divr;Ω ďC hk!}τ}k,r;Ω ` }divpτq}k,r ;Ω),
pAPu
hqthere exists a positive constant C, independent of h, such that for each kP r0, ℓ `1s, and for
each vPWk,rpΩq, there holds
distpv,M1,hq:“inf
vhPM1,h
}v´vh}0,r;Ω ďC hk}v}k,r;Ω ,
32

pAPw
hqthere exists a positive constant C, independent of h, such that for each kP r1, ℓ `1s, and for
each zPWk,rpΩq, with divpzq P Wk,rpΩq, there holds
distpz,X2,hq:“inf
zhPX2,h
}z´zh}r,divr;Ω ďC hk!}z}k,r;Ω ` }divpzq}k,r ;Ω),
pAPp
hqthere exists a positive constant C, independent of h, such that for each kP r0, ℓ `1s, and for
each qPWk,rpΩq, there holds
distpq, M1,hq:“inf
qhPM1,h
}q´qh}0,r;Ω ďC hk}q}k,r;Ω ,
pAPθ
hqthere exists a positive constant C, independent of h, such that for each kP r0, ℓ `1s, and for
each ϑPWk,ρpΩq, there holds
distpϑ, H1,hq:“inf
ϑhPH1,h
}ϑ´ϑh}0,ρ;Ω ďC hk}ϑ}k,ρ;Ω ,
pAPr
t
hqthere exists a positive constant C, independent of h, such that for each kP r0, ℓ `1s, and for
each r
sPHkpΩq, there holds
distpr
s,H2,hq:“inf
r
shPH2,h
}r
s´r
sh}0,ΩďC hk}r
s}k,Ω,
pAPr
σ
hqthere exists a positive constant C, independent of h, such that for each kP r1, ℓ `1s, and for
each r
τPHkpΩq, with divpr
τq P Wk,ϱpΩq, there holds
distpr
τ,Qhq:“inf
r
τhPQh
}r
τ´r
τh}divϱ;Ω ďC hk!}r
τ}k,Ω` }divpr
τq}k,ϱ;Ω).
The rates of convergence of (5.2) are then stated as follows.
Theorem 5.12. Let `pρ,uq,pw, pq,p
θ, r
σq˘P pX2ˆM1qˆ pX2ˆM1qˆ pHˆQq, with pρ, θq P Wpδq, be
the unique solution of (3.34), and let `pρh,uhq,pwh, phq,p
θh,r
σhq˘P pX2,h ˆM1,hq ˆ pX2,h ˆM1,h q ˆ
pHhˆQhq, with pρh, θhq P Whpδq, be a solution of (5.2), which is guaranteed by Theorems 4.11 and
5.10, respectively. Assume the hypotheses of Theorem 5.11 and that there exists kP r1, ℓ `1s, such
that ρPWk,r pΩq X Hr
0pdivr; Ωq,divpρq P Wk,r pΩq,uPWk,r pΩq,wPWk,r pΩq,divpwq P Wk,rpΩq,
pPWk,r pΩq,θPWk,ρpΩq,r
tPHkpΩq,r
σPHkpΩq, and divpr
σq P Wk,ϱpΩq. Then, there exists a positive
constant C, independent of h, such that
}pρ,uq´pρh,uhq}X2ˆM1` }pω, pq´pωh, phq}X2ˆM1` }p
θ, r
σq´p
θh,r
σhq}HˆQ
ďC hk!}ρ}k,r;Ω ` }divpρq}k,r;Ω ` }u}k,r;Ω ` }w}k,r;Ω ` }divpwq}k,r;Ω
` }p}k,r;Ω ` }θ}k,ρ;Ω ` }r
t}k,Ω` }r
σ}k,Ω` }divpr
σq}k,ϱ;Ω).
(5.46)
Proof. It follows straightforwardly from Theorem 5.11 and the above approximation properties.
6 Numerical examples
In this final section we present two sets of computational tests, first the verification of convergence
with respect to manufactured solutions in 2D and 3D, and an application example pertaining to the
33

flow through a deformable porous channel with obstacles and temperature gradient. In all cases we
take the following indexes (according to (3.27), valid for both 2D and 3D) r“3, s“3
2,ρ“6, and
ϱ“6
5. The numerical realization has been done using the finite element library FEniCS [1], selecting
Newton–Raphson as nonlinear solver, with an incremental relative tolerance of 10´8. The linear solves
are done with the direct method MUMPS.
6.1 Example 1: convergence verification
The error history (investigating the error decay with respect to mesh refinement – in a sequence of
successively refined regular grids) is done comparing approximate and closed-form exact solutions
defined on the unit square domain Ω “ p0,1q2. The mixed variables, forcing and source terms for the
balance equations, and non-homogenous boundary data are taken in such a way that the manufactured
primal unknowns are
upx, yq “ 1
10 ˆsinpπxyq
cospπxqcospπyq˙, ppx, yq “ sinpπxqsinpπyq, θpx, yq “ cospxqexpp´x´yq.
The model constants assume the following simple values: µ“1, λ “1, κ “1, α “1, β “1,
χ“1, η “1, whereas the stress-assisted diffusion coefficient is
Dpσq “ D0`D1expp´trpσ2qq ,with D0“0.1 and D1“0.01 .(6.1)
The error history associated with the proposed mixed finite element method on a sequence of
successively refined partitions of the domain, are collected in Table 6.1. Absolute errors are computed
for each variable in the following way
epuq“}u´uh}0,r;Ω, eppq“}p´ph}0,r;Ω , epθq“}θ´θh}0,ρ;Ω, epρq“}ρ´ρh}r,divr;Ω ,
epwq“}w´wh}r,divr;Ω, epr
tq“}r
t´r
th}0,Ω, epr
σq“}r
σ´r
σh}divϱ;Ω,
and we also tabulate rates of error decay computed as rp¨q “ logpep¨q{˜ep¨qqrlogph{˜
hqs´1, where e, ˜e
denote errors generated on two consecutive meshes of sizes hand ˜
h, respectively. All results indicate
optimal convergence of Ophk`1qin all fields and for the two tested polynomial degrees, which coincides
with the theoretical result proposed in Theorem 5.12. For this test we have also tabulated the loss
of momentum and mass conservation by taking the ℓ8norm of the corresponding residuals projected
into the discrete spaces for displacement and pressure. More precisely, letting Pkand Pkbe the
L2pΩq-type and L2pΩq-type orthogonal projectors, respectively, onto the scalar and vector piecewise
polynomials of degree ďk, we set
momh:“ }Pkrdivpσhq ` fs}ℓ8,massh:“ }Pkrc1pλqph´divpwhq ` c3pλqθh`c2pλqtrpρhq ´ fs}ℓ8,
which, according to the second and fourth equations of (5.2), are essentially zero at machine precision.
The table also reports that a maximum of three iterations are needed by the Newton–Raphson method
to reach a tolerance (either absolute or relative) of 10´8on the residual. Sample approximate solutions
for all fields, obtained with the method using k“0, are plotted in Figure 6.1.
The convergence tests are also done in 3D, taking Ω “ p0,1q3, the same model parameters as in
the 2D case, and using the following manufactured primal solutions
upx, y, zq “ 1
10 ¨
˝
sinpπxyzq
cospπxqcospπyqcospπzq
sinpπxqsinpπyqsinpπzq˛
‚, ppx, y, zq “ sinpπxqsinpπyqsinpπzq,
θpx, y, zq “ cospxyqexpp´x´y´zq.
We report on the lowest-order case in Table 6.2 and Figure 6.2, allowing us to draw the same conclusions
as in the 2D case.
34

Primal unknowns and discrete conservation
DoFs h epuqrpuqeppqrppqepθqrpθqmomhmassh
Errors and convergence rates for k“0
113 0.707 4.79e-02 ‹2.78e-01 ‹1.59e-01 ‹4.44e-16 2.11e-15
417 0.354 2.23e-02 1.11 1.50e-01 0.89 8.15e-02 0.96 1.52e-15 4.77e-15
1601 0.177 1.04e-02 1.10 7.65e-02 0.98 4.11e-02 0.99 5.12e-15 9.21e-15
6273 0.088 5.05e-03 1.04 3.84e-02 0.99 2.06e-02 1.00 1.48e-13 2.09e-14
24833 0.044 2.50e-03 1.01 1.92e-02 1.00 1.03e-02 1.00 2.14e-12 5.94e-13
98817 0.022 1.25e-03 1.00 9.61e-03 1.00 5.16e-03 1.00 1.26e-12 2.38e-13
Errors and convergence rates for k“1
337 0.707 1.22e-02 ‹8.91e-02 ‹1.48e-02 ‹6.46e-15 7.49e-15
1281 0.354 3.02e-03 2.02 2.29e-02 1.96 3.64e-03 2.02 1.17e-14 3.50e-14
4993 0.177 7.48e-04 2.01 5.83e-03 1.98 9.13e-04 1.99 3.08e-14 5.20e-14
19713 0.088 1.86e-04 2.01 1.46e-03 1.99 2.29e-04 2.00 7.41e-14 1.51e-13
78337 0.044 4.65e-05 2.00 3.66e-04 2.00 5.72e-05 2.00 1.49e-13 3.46e-13
312321 0.022 1.16e-05 2.00 9.16e-05 2.00 1.43e-05 2.00 1.66e-12 1.02e-12
Mixed unknowns and iteration count
DoFs h epρqrpρqepwqrpwqepr
tqrpr
tqepr
σqrpr
σqiter
Errors and convergence rates for k“0
113 0.707 2.14e+00 ‹6.50e+00 ‹1.50e-01 ‹1.08e-01 ‹3
417 0.354 1.16e+00 0.88 3.55e+00 0.87 7.38e-02 1.02 5.06e-02 1.09 3
1601 0.177 5.98e-01 0.96 1.81e+00 0.98 3.70e-02 1.00 2.81e-02 0.85 3
6273 0.088 3.01e-01 0.99 9.07e-01 0.99 1.86e-02 0.99 1.48e-02 0.93 3
24833 0.044 1.51e-01 1.00 4.54e-01 1.00 9.31e-03 1.00 7.48e-03 0.98 3
98817 0.022 7.54e-02 1.00 2.27e-01 1.00 4.65e-03 1.00 3.75e-03 1.00 3
Errors and convergence rates for k“1
337 0.707 6.93e-01 ‹1.99e+00 ‹3.31e-02 ‹6.14e-02 ‹3
1281 0.354 2.00e-01 1.79 5.12e-01 1.96 4.77e-03 2.79 1.44e-02 2.10 3
4993 0.177 5.21e-02 1.94 1.30e-01 1.98 1.27e-03 1.91 4.65e-03 1.63 3
19713 0.088 1.32e-02 1.98 3.26e-02 1.99 3.39e-04 1.91 1.33e-03 1.80 3
78337 0.044 3.31e-03 1.99 8.16e-03 2.00 8.82e-05 1.94 3.31e-04 2.01 3
312321 0.022 8.29e-04 2.00 2.04e-03 2.00 2.23e-05 1.98 8.33e-05 1.99 3
Table 6.1: Example 1 (2D). Error history for the primal unknowns together with discrete approxima-
tion of momentum and mass conservation (top table) and convergence of mixed unknowns together
with Newton–Raphson iteration count with respect to mesh refinement (bottom table). The symbol
‹indicates that no convergence rate is computed at that refinement level.
6.2 Example 2: injection of fluid in a deformable porous channel
To conclude this section, we investigate the flow patterns of infiltration of a poroelastic channel having
an irregular array of eight circular cylinders that are maintained at a low temperature. The problem
setup mimics the behaviour of sponge-like materials or soils in the presence of macro-pores, for example
[18,32]. The undeformed body occupies the rectangular domain Ω “ p0,1.6qˆp0,1q(in m2), which
we discretize into an unstructured mesh of 55450 triangles.
We consider a simple time-dependent version of the model (2.1), where only the energy balance
equation (2.1c) is modified to have Btθ. We use a backward Euler discretization in time, with constant
time step ∆t“1 (in s) and an initial temperature of 10 degrees. In addition, the boundary conditions
35

Figure 6.1: Example 1 (2D). Verification of convergence with respect to manufactured solutions.
Approximate primal (top) and mixed (bottom) unknowns computed using the lowest-order scheme,
and portrayed in the deformed configuration (the outline of the undeformed domain is also shown for
reference).
Primal unknowns and discrete conservation
DoFs h epuqrpuqeppqrppqepθqrpθqmomhmassh
139 1.732 7.10e-02 ‹3.27e-01 ‹2.18e-01 ‹6.45e-16 4.00e-15
985 0.866 3.82e-02 0.89 2.23e-01 0.55 1.33e-01 0.71 1.16e-15 6.41e-15
7393 0.433 1.91e-02 1.00 1.17e-01 0.94 7.12e-02 0.90 2.72e-15 1.08e-14
57217 0.217 9.35e-03 1.03 5.96e-02 0.97 3.63e-02 0.97 7.76e-15 2.07e-14
450049 0.108 4.63e-03 1.01 3.00e-02 0.99 1.82e-02 0.99 1.99e-14 3.79e-14
Mixed unknowns and iteration count
DoFs h epρqrpρqepwqrpwqepr
tqrpr
tqepr
σqrpr
σqiter
139 1.732 3.76e+00 ‹1.06e+01 ‹2.65e-01 ‹7.99e-02 ‹3
985 0.866 2.17e+00 0.79 7.55e+00 0.49 1.29e-01 1.04 4.67e-02 0.78 3
7393 0.433 1.16e+00 0.90 4.01e+00 0.91 6.56e-02 0.98 2.90e-02 0.69 3
57217 0.217 5.90e-01 0.97 2.05e+00 0.97 3.27e-02 1.00 1.65e-02 0.81 3
450049 0.108 2.96e-01 0.99 1.03e+00 0.99 1.63e-02 1.00 8.64e-03 0.94 3
Table 6.2: Example 1 (3D). Error history for the primal unknowns together with discrete approxima-
tion of momentum and mass conservation (top table) and convergence of mixed unknowns together
with Newton–Raphson iteration count with respect to mesh refinement (bottom table). The symbol
‹indicates that no convergence rate is computed at that refinement level.
are of mixed type and do not coincide exactly with those analyzed in the manuscript. The left segment
is considered an inflow boundary where we set zero displacements (as a natural boundary condition),
36

Figure 6.2: Example 1 (3D). Verification of convergence with respect to manufactured solutions.
Approximate primal (top) and mixed (bottom) unknowns computed using the lowest-order scheme,
and portrayed in the deformed configuration (the outline of the undeformed domain is also shown for
reference).
a time-dependent parabolic profile as inflow of filtration flux (as an essential boundary condition), and
a quadratic temperature profile (natural boundary condition)
u“0,w¨ν“t
20atanpyr1´ysq m/s, θ “ ´74y2`91y`3pin ˝Cqon Γin;
on the horizontal walls we approximate a zero-traction boundary condition with a zero normal pseu-
dostress condition (imposed essentially), zero normal flux (essential), and a hot temperature on the
top of the channel and cold on the bottom (natural boundary conditions)
ρν “0,w¨ν“0, θ “θD,on Γwall ,
(where θDis 3 degrees on the bottom and 20 degrees on the top); on the holes we impose
u“0,w¨ν“0, θ “3˝C,on Γcyl ;
and the boundary conditions are completed by prescribing zero traction (approximated by a zero
normal pseudostress), a vanishing pressure (natural boundary condition), and a zero thermal flux on
37

∥u∥
p
θ
∥w∥
∥
e
t∥
∥e
σ∥
Figure 6.3: Example 2. Fluid injection using Biot–heat equations on a deformable channel with an
array of cylinders, plotted on the undeformed configuration at time t“50 s. Approximate solutions
computed with a second-order method.
the outlet region (essentially imposed)
ρν “0, p “0,r
σ¨ν“0,on Γout.
We do not consider external volume forces nor fluid sources, therefore f“0,f“g“0, the stress-
assisted diffusion term is as in Example 1 (cf. (6.1)) with D0“10´3and D1“10´4, and the
remaining physical parameters are all constant and assuming the values
µ“210 Pa, λ “1800 Pa, η “10´3Pa s, κ “10´5m2, α “0.9, β “1.5, χ “10´2Pa.
The simulation runs until t“50s. The numerical solutions obtained with a second-order scheme
(setting k“1, for which the method consists of 667928 DoFs) are portrayed in Figure 6.3, showing
snapshots of the deformed poroelastic region, filtration flux, and all other field variables at the final
time. The expected injection patterns are seen in the flux plot, as well as the progressive heating of
the fluid near the top plate.
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41

Centro de Investigaci´on en Ingenier´ıa Matem´atica (CI2MA)
PRE-PUBLICACIONES 2023 - 2024
2023-22 Franz Chouly:A short journey into the realm of numerical methods for contact in
elastodynamics
2023-23 St´
ephane P. A. Bordas, Marek Bucki, Huu Phuoc Bui, Franz Chouly,
Michel Duprez, Arnaud Lejeune, Pierre-Yves Rohan:Automatic mesh re-
finement for soft tissue
2023-24 Mauricio Sep´
ulveda, Nicol´
as Torres, Luis M. Villada:Well-posedness and
numerical analysis of an elapsed time model with strongly coupled neural networks
2023-25 Franz Chouly, Patrick Hild, Yves Renard:Lagrangian and Nitsche methods
for frictional contact
2023-26 Sergio Caucao, Gabriel N. Gatica, Juan P. Ortega:A three-field mixed
finite element method for the convective Brinkman–Forchheimer problem with varying
porosity
2023-27 Raimund B¨
urger, Yessennia Mart
´
ınez, Luis M. Villada:Front tracking and
parameter identification for a conservation law with a space-dependent coefficient mo-
deling granular segregation
2023-28 Marie Haghebaert, Beatrice Laroche, Mauricio Sep´
ulveda:Study of the
numerical method for an inverse problem of a simplified intestinal crypt
2023-29 Rodolfo Araya, Fabrice Jaillet, Diego Paredes, Frederic Valentin:Gen-
eralizing the Multiscale Hybrid-Mixed Method for Reactive-Advective-Diffusive Equa-
tions
2023-30 Jessika Cama˜
no, Ricardo Oyarz´
ua, Miguel Ser´
on, Manuel Solano:A
mass conservative finite element method for a nonisothermal Navier-Stokes/Darcy cou-
pled system
2023-31 Franz Chouly, Hao Huang, Nicol´
as Pignet:HHT-αand TR-BDF2 schemes
for Nitsche-based discrete dynamic contact
2024-01 Sergio Caucao, Gabriel N. Gatica, Saulo Medrado, Yuri D. Sobral:
Nonlinear twofold saddle point-based mixed finite element methods for a regularized
µ(I)-rheology model of granular materials
2024-02 Julio Careaga, Gabriel N. Gatica, Cristian Inzunza, Ricardo Ruiz-Baier:
New Banach spaces-based mixed finite element methods for the coupled poroelasticity
and heat equations
Para obtener copias de las Pre-Publicaciones, escribir o llamar a: Director, Centro de
Investigaci´
on en Ingenier
´
ıa Matem´
atica, Universidad de Concepci´
on, Casilla
160-C, Concepci´
on, Chile, Tel.: 41-2661324, o bien, visitar la p´agina web del centro:
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Centro de Investigaci´
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Ingenier
´
ıa Matem´
atica (CI2MA)
Universidad de Concepci´on
Casilla 160-C, Concepci´on, Chile
Tel.: 56-41-2661324/2661554/2661316
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