# A new framework for the stability analysis of perturbed saddle-point
problems and applications in poromechanics

Qingguo Hong  Email*<EMAIL_ADDRESS>
Department of Mathematics, Pennsylvania State University, University Park, PA
16802, USA  ,  Johannes Kraus  Email<EMAIL_ADDRESS>
Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9,
Essen 45127, Germany  ,  Maria Lymbery  Email<EMAIL_ADDRESS>
Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9,
Essen 45127, Germany  and  Fadi Philo  Email<EMAIL_ADDRESS>
Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9,
Essen 45127, Germany

###### Abstract.

This paper provides a new abstract stability result for perturbed saddle-point
problems which is based on a proper norm fitting. We derive the stability
condition according to Babuška's theory from a small inf-sup condition,
similar to the famous Ladyzhenskaya-Babuška-Brezzi (LBB) condition, and the
other standard assumptions in Brezzi's theory under the resulting combined
norm. The proposed framework allows to split the norms into proper seminorms
and not only results in simpler (shorter) proofs of many stability results but
also guides the construction of parameter robust norm-equivalent
preconditioners. These benefits are demonstrated with several examples arising
from different formulations of Biot's model of consolidation.

###### Key words and phrases:

fitted norms, stability analysis, saddle-point problems, LBB condition,
poromechanics, Biot's model

##  1\. Introduction

Saddle-point problems (SPPs) arise in various areas of computational science
and engineering ranging from computational fluid dynamics [22, 48, 23],
elasticity [2, 20, 13], and electromagnetics [39, 10] to computational finance
[31]. Moreover, SPPs play a vital role in the context of image reconstruction
[25], model order reduction [47], constrained optimization [21], optimal
control [6], and parameter identification [16], to mention only a few but
important applications.

In the mathematical modeling of multiphysics phenomena described by (initial-)
boundary-value problems for systems of partial differential equations, SPPs
often naturally arise and are frequently posed in a variational formulation.
Mixed finite element methods and other discretization techniques can and have
been successfully used for their discretization and numerical solution, see,
e.g. [15, 19, 7, 10] and the references therein.

The pioneering works laying the foundations of the solution theory for SPPs
have been conducted by Jindřich Nečas, Olga Ladyzhenskaya, Ivo Babuška, and
Franco Brezzi [42, 33, 3, 14], see also the contributions [4, 32].

Designing and analyzing discretizations and solvers for SPPs require a careful
study of the mapping properties of the underlying operators. Of particular
interest are their continuity and stability, which not only guarantee the
well-posedness of (continuous and discrete) mathematical models but also
provide the basis for error estimates and a convergence analysis of iterative
methods and preconditioners, see, e.g. [37, 19, 38, 10], for a review see also
[7].

Saddle-point problems/systems are of a two-by-two block form and characterized
by an operator/matrix of the form

(1) $\mathcal{A}=\begin{pmatrix}A&B_{1}^{T}\\\ B_{2}&-C\end{pmatrix}$

where $\displaystyle A$ and $\displaystyle C$ denote positive semidefinite
operators/matrices and $\displaystyle B^{T}$ the adjoint/transpose of an
operator/matrix $\displaystyle B$. We consider the symmetric case in this
paper where $\displaystyle A^{T}=A$, $\displaystyle C^{T}=C$, and
$\displaystyle B_{1}=B_{2}=B$. Problems in which $\displaystyle C\neq 0$ are
often referred to as perturbed saddle-point problems. They are in the focus of
this paper.

More general SPPs in which $\displaystyle A$ (and $\displaystyle C$) are
allowed to be nonsymmetric and $\displaystyle B_{1}\neq B_{2}$ have also been
studied by many authors, see, e.g., [43, 15, 10], and the references therein.
Their analysis, in general is more complicated and is mostly done following
the monolithic approach, i.e., imposing conditions on
$\displaystyle\mathcal{A}$ rather than on $\displaystyle A$, $\displaystyle
B_{1}$, $\displaystyle B_{2}$, and $\displaystyle C$ separately.

Our work is motivated by the stability analysis of variational problems
occurring in poromechanics (cf. [17]), a subarea of continuum mechanics which
originates from the early works of Terzaghi and Biot [49, 8]. Various
formulations of Biot's consolidation model have been considered and analyzed
since it had been introduced in [8, 9], including two-field ([40, 41]), three-
field ([45, 46, 44, 34, 26]), and four-field-formulations ([51, 35, 30]), for
generalizations to several fluid networks as considered in multiple network
poroelastic theory (MPET), see also [5, 24, 27, 28, 29, 36, 50].

Although they typically relate more than two physical fields, or quantities of
interest (except for the two-field formulation of Biot's model), the
variational problems arising from the above-mentioned formulations–subject to
a proper grouping or rather aggregation of variables–result in symmetric two-
by-two block systems of saddle point form characterized by a self-adjoint
operator $\displaystyle\mathcal{A}$.

The abstract framework presented in the next section of this paper applies to
such saddle-point operators. After introducing some notation, we recall the
classical stability results of Babuška and Brezzi for classical (unperturbed)
SPPs. Next, we focus on perturbed (symmetric) SPPs, initially summarizing some
of the additional conditions which, together with the Ladyzhenskaya-Babuška-
Brezzi (LBB) condition (small inf-sup condition), imply the necessary and
sufficient stability condition of Babuška (big inf-sup condition). Our main
theoretical result then follows in Section 2.3 where we propose a
generalization of the classical Brezzi conditions for the analysis of
perturbed SPPs with $\displaystyle C\neq 0$. These new conditions imply the
Babuška condition. The fitted norms on which they are based provide a
constructive tool for designing norm-equivalent preconditioners.

This paper does not discuss discretizations and discrete variants of inf-sup
conditions, although the proposed framework directly translates to discrete
settings where it would also allow for shorter and simpler proofs of the well-
posedness of discrete models and error-estimates for stable discretizations.

##  2\. Abstract framework

###  2.1. Notation and problem formulation

Consider two Hilbert spaces $\displaystyle V$ and $\displaystyle Q$ equipped
with the norms $\displaystyle\|\cdot\|_{V}$ and $\displaystyle\|\cdot\|_{Q}$
induced by the scalar products $\displaystyle(\cdot,\cdot)_{V}$ and
$\displaystyle(\cdot,\cdot)_{Q}$, respectively. We denote their product space
by $\displaystyle Y:=V\times Q$ and endow it with the norm
$\displaystyle\|\cdot\|_{Y}$ defined by

(2)
$\|y\|_{Y}^{2}=(y,y)_{Y}=(v,v)_{V}+(q,q)_{Q}=\|v\|_{V}^{2}+\|q\|_{Q}^{2}\qquad\forall
y=(v;q):=\begin{pmatrix}v\\\ q\end{pmatrix}\in Y.$

Next, we introduce an abstract bilinear form
$\displaystyle\mathcal{A}((\cdot\,;\cdot),(\cdot\,;\cdot))$ on $\displaystyle
Y\times Y$ defined by

(3) $\mathcal{A}((u;p),(v;q)):=a(u,v)+b(v,p)+b(u,q)-c(p,q)$

for some symmetric positive semidefinite bilinear forms $\displaystyle
a(\cdot,\cdot)$ on $\displaystyle V\times V$, $\displaystyle c(\cdot,\cdot)$
on $\displaystyle Q\times Q$, i.e.,

(4) $\displaystyle\displaystyle a(u,v)$
$\displaystyle\displaystyle=a(v,u)\qquad\forall u,v\in V,$ (5)
$\displaystyle\displaystyle a(v,v)$ $\displaystyle\displaystyle\geq
0\qquad\forall v\in V,$ (6) $\displaystyle\displaystyle c(p,q)$
$\displaystyle\displaystyle=c(q,p)\qquad\forall p,q\in Q,$ (7)
$\displaystyle\displaystyle c(q,q)$ $\displaystyle\displaystyle\geq
0\qquad\forall q\in Q.$

and a bilinear form $\displaystyle b(\cdot,\cdot)$ on $\displaystyle V\times
Q$.

We assume that $\displaystyle a(\cdot,\cdot)$, $\displaystyle b(\cdot,\cdot)$
and $\displaystyle c(\cdot,\cdot)$ are continuous with respect to the norms
$\displaystyle\|\cdot\|_{V}$ and $\displaystyle\|\cdot\|_{Q}$, i.e.,

(8) $\displaystyle\displaystyle a(u,v)$
$\displaystyle\displaystyle\leq\bar{C}_{a}\|u\|_{V}\|v\|_{V}\qquad\forall
u,v\in V,$ (9) $\displaystyle\displaystyle b(v,q)$
$\displaystyle\displaystyle\leq\bar{C}_{b}\|v\|_{V}\|q\|_{Q}\qquad\forall v\in
V,\forall q\in Q,$ (10) $\displaystyle\displaystyle c(p,q)$
$\displaystyle\displaystyle\leq\bar{C}_{c}\|p\|_{Q}\|q\|_{Q}\qquad\forall
p,q\in Q.$

Then each of these bilinear forms defines a bounded linear operator as
follows:

(11a) $\displaystyle\displaystyle A:V\rightarrow V^{\prime}:\,\,$
$\displaystyle\displaystyle\langle Au,v\rangle_{V^{\prime}\times
V}=a(u,v),\qquad\forall u,v\in V,$ (11b) $\displaystyle\displaystyle
C:Q\rightarrow Q^{\prime}:\,\,$ $\displaystyle\displaystyle\langle
Cp,q\rangle_{Q^{\prime}\times Q}=c(p,q),\qquad\forall p,q\in Q,$ (11c)
$\displaystyle\displaystyle B:V\rightarrow Q^{\prime}:\,\,$
$\displaystyle\displaystyle\langle Bv,q\rangle_{Q^{\prime}\times
Q}=b(v,q),\qquad\forall v\in V,\forall q\in Q,$ (11d)
$\displaystyle\displaystyle B^{T}:Q\rightarrow V^{\prime}:\,\,$
$\displaystyle\displaystyle\langle v,B^{T}q\rangle_{V\times
V^{\prime}}=b(v,q),\qquad\forall v\in V,\forall q\in Q.$

Here $\displaystyle V^{\prime}$ and $\displaystyle Q^{\prime}$ denote the dual
spaces of $\displaystyle V$ and $\displaystyle Q$, respectively, and
$\displaystyle\langle\cdot,\cdot\rangle$ the corresponding duality pairing.

Associated with the bilinear form $\displaystyle\mathcal{A}$ defined in (3) we
consider the following abstract perturbed saddle-point problem

(12) $\mathcal{A}((u;p),(v;q))=\mathcal{F}((v;q))\qquad\forall v\in V,\forall
q\in Q$

which can also be written as

$\displaystyle\mathcal{A}(x,y)=\mathcal{F}(y)\qquad\forall y\in Y,$

thereby using the definitions $\displaystyle x=(u;p)$ and $\displaystyle
y=(v;q)$, or, in operator form

(13) $\mathcal{A}x=\mathcal{F}$

where

(14) $\displaystyle\displaystyle\mathcal{A}:Y\rightarrow Y^{\prime}:\,\,$
$\displaystyle\displaystyle\langle\mathcal{A}x,y\rangle_{Y^{\prime}\times
Y}=\mathcal{A}(x,y),\qquad\forall x,y\in Y$

and $\displaystyle\mathcal{F}\in Y^{\prime}$, i.e.,
$\displaystyle\mathcal{F}:Y\rightarrow\mathbb{R}:\mathcal{F}(y)=\langle\mathcal{F},y\rangle_{Y^{\prime}\times
Y}$ for all $\displaystyle y\in Y.$

The operator $\displaystyle\mathcal{A}$ can also be represented in block form
by

(15) $\mathcal{A}=\begin{pmatrix}A&B^{T}\\\ B&-C\end{pmatrix}.$

Problem (12) (and (13)) is called a perturbed saddle-point problem (in
operator form) when $\displaystyle c(\cdot,\cdot)\not\equiv 0$ and a classical
saddle-point problem in the case $\displaystyle c(\cdot,\cdot)\equiv 0$.

###  2.2. Babuška's and Brezzi's conditions for stability of saddle-point
problems

As is well-known from [3], the abstract variational problem (12) is well-posed
under the following necessary and sufficient conditions (i) and (ii) given in
the following theorem.

######  Theorem 1 (Babuška [3]).

Let $\displaystyle\mathcal{F}\in Y^{\prime}$ be a bounded linear functional.
Then the saddle-point problem (12) is well-posed if and only if there exist
positive constants $\displaystyle\bar{C}$ and $\displaystyle\alpha$ for which
the conditions

(16) $\mathcal{A}(x,y)\leq\bar{C}\|x\|_{Y}\|y\|_{Y}\qquad\forall x,y\in Y,$
(17) $\inf_{x\in Y}\sup_{y\in
Y}\frac{\mathcal{A}(x,y)}{\|x\|_{Y}\|y\|_{Y}}\geq\underline{\alpha}>0$

hold. The solution $\displaystyle x$ then satisfies the stability estimate

$\|x\|_{Y}\leq\frac{1}{\underline{\alpha}}\sup_{y\in
Y}\frac{\mathcal{F}(y)}{\|y\|_{Y}}=:\frac{1}{\underline{\alpha}}\|\mathcal{F}\|_{Y^{\prime}}.$

######  Remark 1.

Estimate (16) ensures continuity, that is, boundedness of the operator
$\displaystyle\mathcal{A}$ from above, whereas (17) is a stability condition,
sometimes referred to as Babuška condition, which grants boundedness of
$\displaystyle\mathcal{A}$ from below.

For the classical saddle-point problem, i.e., $\displaystyle
c(\cdot,\cdot)\equiv 0$, the following theorem which we formulate under the
conditions that $\displaystyle a(\cdot,\cdot)$ is symmetric positive
semidefinite and

(18) ${\rm Ker}(B^{T}):=\\{q\in Q:b(v,q)=0\,\,\forall v\in V\\}=\emptyset$

has been proven in [14], see also [15, 10].

######  Theorem 2.

[Brezzi [14]] Assume that the bilinear forms $\displaystyle a(\cdot,\cdot)$
and $\displaystyle b(\cdot,\cdot)$ are continuous on $\displaystyle V\times V$
and on $\displaystyle V\times Q$, respectively, $\displaystyle a(\cdot,\cdot)$
is symmetric positive semidefinite, and also that

(19) $a(v,v)\geq\underline{C}_{a}\|v\|_{V}\qquad\forall v\in{\rm Ker}(B),$
(20) $\inf_{q\in Q}\sup_{v\in V}\frac{b(v,q)}{\|v\|_{V}\|q\|_{Q}}\geq\beta>0,$

hold. Then the classical saddle-point problem (problem (12) with
$\displaystyle c(\cdot,\cdot)\equiv 0$) is well-posed.

######  Remark 2.

Note that if $\displaystyle{\rm Ker}(B^{T})\neq\emptyset$, the statement of
the above theorem (Theorem 2) remains valid if we identify any two elements
$\displaystyle q_{1}$, $\displaystyle q_{2}$ for which $\displaystyle
q_{0}:=q_{1}-q_{2}$ is an element of $\displaystyle{\rm{Ker}}(B^{T})$, i.e.,
replacing the space $\displaystyle Q$ with the quotient space $\displaystyle
Q/{\rm{Ker}}(B^{T})$ and also the norm $\displaystyle\|\cdot\|_{Q}$ with
$\displaystyle\|\cdot\|_{Q/{\rm{Ker}}(B^{T})}$, the latter being defined by

$\displaystyle\|q\|_{Q/{\rm{Ker}}(B^{T})}=\inf_{q_{0}\in{\rm
Ker}(B^{T})}\|q+q_{0}\|_{Q}.$

In this case, the solution $\displaystyle p$ is only unique up to an arbitrary
element $\displaystyle p_{0}\in{\rm Ker}(B^{T})$.

For the classical saddle-point problem Brezzi's stability condition (20) and
the continuity of $\displaystyle a(\cdot,\cdot)$ imply Babuška's stability
condition (17), see [18], where it has also been shown that from (17) it
follows (20) and the inf-sup condition for $\displaystyle a(\cdot,\cdot)$ in
the kernel of $\displaystyle B$, the latter being equivalent to the coercivity
estimate (19) if $\displaystyle a(\cdot,\cdot)$ is symmetric positive
semidefinite.

Obviously, the stability condition (17) directly applies to perturbed saddle-
point problems, a reason why they can be studied using Babuška's theory.
However, conditions (19) and (20) together with the continuity of
$\displaystyle a(\cdot,\cdot)$, $\displaystyle b(\cdot,\cdot)$ and
$\displaystyle c(\cdot,\cdot)$ in general are not sufficient to guarantee the
stability condition (17) when $\displaystyle c(\cdot,\cdot)\not\equiv 0$.
Additional conditions to ensure (17) have been studied, for example in [15,
12, 10].

In [15] it has been shown that a condition on the kernel of $\displaystyle
B^{T}$ can be used as an additional assumption to ensure well-posedness of the
perturbed saddle-point problem, that is, in particular, for Babuška's inf-sup
condition (17) to hold. This condition is expressed in terms of the following
auxiliary problem

(21) $\epsilon(p_{0},q)_{Q}+c(p_{0},q)=-c(p^{\perp},q),\qquad\forall q\in{\rm
Ker}(B^{T})$

and requires the following general assumption:

######  Assumption 1.

There exists a $\displaystyle\gamma_{0}>0$ such that for every $\displaystyle
p^{\perp}\in({\rm Ker}(B^{T}))^{\perp}$ and for every
$\displaystyle\epsilon>0$ it holds that the solution $\displaystyle
p_{0}\in{\rm Ker}(B^{T})$ of (21) is bounded by
$\displaystyle\gamma_{0}\|p_{0}\|_{Q}\leq\|p^{\perp}\|_{Q}$.

The theorem then reads as follows:

######  Theorem 3 (Brezzi and Fortin [15]).

Assume that $\displaystyle a(\cdot,\cdot)$, $\displaystyle b(\cdot,\cdot)$ and
$\displaystyle c(\cdot,\cdot)$ are continuous bilinear forms on $\displaystyle
V\times V$, on $\displaystyle V\times Q$, and on $\displaystyle Q\times Q$,
respectively. Further assume that $\displaystyle a(\cdot,\cdot)$ and
$\displaystyle c(\cdot,\cdot)$ are symmetric positive semidefinite. Finally,
let (19), (20) (conditions (i) and (ii) from Theorem 2) and Assumption 1 be
satisfied. Then for every $\displaystyle f\in V^{\prime}$ and every
$\displaystyle g\in{\rm Im}(B)$ problem (12) with $\displaystyle\mathcal{A}$
as defined in (3) and $\displaystyle\mathcal{F}(y)=\langle
f,v\rangle_{V^{\prime}\times V}+\langle g,q\rangle_{Q^{\prime}\times Q}$ has a
unique solution $\displaystyle x=(u;p)$ in $\displaystyle Y=V\times Q/M$ where

$M={\rm Ker}(B^{T})\cap{\rm Ker}(C).$

Moreover, the estimate

$\|u\|_{V}+\|p\|_{Q/{\rm Ker}(B^{T})}\leq
K(\|f\|_{V^{\prime}}+\|g\|_{Q^{\prime}})$

holds with a constant $\displaystyle K$ only depending on
$\displaystyle\bar{C}_{a}$, $\displaystyle\bar{C}_{c}$,
$\displaystyle\underline{C}_{a}$, $\displaystyle\beta$ and
$\displaystyle\gamma_{0}$.

######  Remark 3.

The result in [15] is more general than Theorem 3 in that it applies also to
non-symmetric but positive semidefinite $\displaystyle a(\cdot,\cdot)$. We are
considering only the case of symmetric positive semidefinite $\displaystyle
a(\cdot,\cdot)$ in this paper.

In order to ensure the boundedness (continuity) of the symmetric positive
semidefinite bilinear form $\displaystyle c(\cdot,\cdot)$ with respect to the
norm $\displaystyle\|\cdot\|_{Q}$, and more generally the boundedness of
$\displaystyle\mathcal{A}(\cdot,\cdot)$ with respect to the combined norm
$\displaystyle\|\cdot\|_{Y}=(\|\cdot\|_{V}^{2}+\|\cdot\|_{Q}^{2})^{1/2}$, it
is natural to include the contribution of $\displaystyle c(\cdot,\cdot)$ in
the norm $\displaystyle\|\cdot\|_{Q}$, e.g., by defining
$\displaystyle\|\cdot\|_{Q}$ via

(22) $\|q\|^{2}_{Q}=|q|_{Q}^{2}+t^{2}c(q,q),\qquad\forall q\in Q,$

for a proper seminorm or norm $\displaystyle|\cdot|_{Q}$ and a parameter
$\displaystyle t\in[0,1]$.

As has been shown in [12] the stability of the perturbed saddle-point problem
then can be proven under Brezzi's conditions for the classical saddle-point
problem and the additional condition

(23) $\inf_{u\in V}\sup_{(v;q)\in V\times
Q}\frac{a(u,v)+b(u,q)}{{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|(v;q)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}}\geq\gamma>0$

where
$\displaystyle{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
is defined by

(24)
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|(v;q)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}^{2}:=\|v\|_{V}^{2}+|q|_{Q}^{2}+t^{2}c(q,q),\qquad
t\in[0,1],$

and provides a specific choice of $\displaystyle\|\cdot\|_{Y}$, i.e.,
$\displaystyle\|\cdot\|_{Y}={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$.
The corresponding theorem then reads as:

######  Theorem 4 (Braess [12]).

Assume that the classical saddle-point problem is stable, i.e., Brezzi's
conditions (19) and (20) are fulfilled. If in addition condition (23) holds
with $\displaystyle\gamma>0$ and we choose $\displaystyle t>0$ in (24) for
$\displaystyle c(\cdot,\cdot)\not\equiv 0$, then the perturbed saddle-point
problem (12) is stable under the norm
$\displaystyle\|\cdot\|_{Y}:={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
and the stability constant $\displaystyle\underline{\alpha}$ in (17) depends
only on $\displaystyle\beta$, $\displaystyle\underline{C}_{a}$ and
$\displaystyle\gamma$ and the choice of $\displaystyle t$.

######  Remark 4.

Note that as it can easily be seen if $\displaystyle a(\cdot,\cdot)$ is
symmetric positive semidefinite, condition (23) is equivalent to the condition
that there exists a constant $\displaystyle\gamma^{\prime}>0$ such that

(25) $\frac{a(u,u)}{\|u\|_{V}}+\sup_{q\in
Q}\frac{b(u,q)}{|q|_{Q}+tc(q,q)}\geq\gamma^{\prime}\|u\|_{V}\qquad\forall u\in
V.$

Moreover, as shown in [12], then (25) is also equivalent to the condition that
there exists a constant $\displaystyle\gamma^{\prime\prime}>0$ such that

(26) $\sup_{(v;q)\in
Y}\frac{\mathcal{A}((u;0),(v;q))}{{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|(v;q)\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}}\geq\gamma^{\prime\prime}\|u\|_{V}\qquad\forall
u\in V.$

Since (23) is an inf-sup condtion for
$\displaystyle(a(\cdot,\cdot)+b(\cdot,\cdot))$ which can be interpreted as a
big inf-sup condition on $\displaystyle\mathcal{A}$ for $\displaystyle p=0$
under the specific norm
$\displaystyle\|\cdot\|_{Y}={\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$,
see (26), Theorem 4 still does not provide us with the desired stability
result in terms of conditions on $\displaystyle a(\cdot,\cdot)$,
$\displaystyle b(\cdot,\cdot)$ and $\displaystyle c(\cdot,\cdot)$ separately.
On the other hand, Theorem 3 requires the solution of the auxiliary problem
(21) on $\displaystyle{\rm Ker}(B^{T})$ for which one has to verify Assumption
1 which, in some situations, is a difficult task.

Our aim is to avoid the latter and still impose Brezzi-type conditions, in
particular a small inf-sup condition on $\displaystyle b(\cdot,\cdot)$. In the
next section, we will prove a theorem (Theorem 5) which ensures the stability
of the perturbed saddle-point problem (12) under conditions which are
equivalent to the conditions in Brezzi’s theorem (Theorem 2) when the
perturbation term vanishes. Moreover, our approach provides a framework suited
for finding norms in which stability can be shown and allows for simplifying
and shortening proofs based on the result of Babuška.

###  2.3. A new framework for the stability analysis of perturbed saddle-point
problems

The key idea for studying and verifying the stability of perturbed saddle-
point problems we follow in this paper is to construct proper norms as part of
an abstract framework which applies to a variational formulation of various
multiphysics models. As we have already observed in the previous subsection, a
norm-splitting of the form (22) is quite natural if the symmetric positive
semidefinite perturbation form $\displaystyle c(\cdot,\cdot)$ is not identical
to zero. For fixed $\displaystyle t>0$, the norm defined in (22) is equivalent
to the norm defined by

(27)
$\|q\|_{Q}^{2}:=|q|_{Q}^{2}+c(q,q)=:\langle\bar{Q}q,q\rangle_{Q^{\prime}\times
Q}$

where $\displaystyle\bar{Q}:Q\rightarrow Q^{\prime}$ is a linear operator,
which follows from the fact that the norm $\displaystyle\|\cdot\|_{Q}$ is
induced by the scalar product $\displaystyle(\cdot,\cdot)_{Q}$ on the Hilbert
space $\displaystyle Q$. Now we introduce the following splitting of the norm
$\displaystyle\|\cdot\|_{V}$ defined by

(28) $\displaystyle\displaystyle\|v\|^{2}_{V}$
$\displaystyle\displaystyle:=|v|^{2}_{V}+|v|^{2}_{b}$

where $\displaystyle|\cdot|_{V}$ is a proper seminorm, which is a norm on
$\displaystyle{\rm Ker}(B)$ satisfying

$\displaystyle|v|_{V}^{2}\eqsim a(v,v)\qquad\forall v\in{\rm Ker}(B)$

and $\displaystyle|\cdot|_{b}$ is defined by

(29) $\displaystyle\displaystyle|v|^{2}_{b}:=\langle
Bv,\bar{Q}^{-1}Bv\rangle_{Q^{\prime}\times Q}=\|Bv\|_{Q^{\prime}}^{2}.$

Here, $\displaystyle\bar{Q}^{-1}:Q^{\prime}\rightarrow Q$ is an isometric
isomorphism since $\displaystyle\bar{Q}$ is an isometric isomorphism (Riesz
isomorphism), i.e.,

$\displaystyle\displaystyle\|Bv\|_{Q^{\prime}}^{2}$
$\displaystyle\displaystyle=\|\bar{Q}^{-1}Bv\|_{Q}^{2}=(\bar{Q}^{-1}Bv,\bar{Q}^{-1}Bv)_{Q}=\langle\bar{Q}\bar{Q}^{-1}Bv,\bar{Q}^{-1}Bv\rangle_{Q^{\prime}\times
Q}=\langle Bv,\bar{Q}^{-1}Bv\rangle_{Q^{\prime}\times Q}.$

######  Remark 5.

Note that both $\displaystyle|\cdot|_{V}$ and $\displaystyle|\cdot|_{b}$ can
be seminorms as long as they add up to a full norm. Likewise, only the sum of
the seminorms $\displaystyle|\cdot|_{Q}$ and $\displaystyle c(\cdot,\cdot)$
has to define a norm. In some particular situations, it is also useful to
identify certain of the involved seminorms with $\displaystyle 0$, in which
case the corresponding splitting becomes a trivial splitting. The splitting
(28) is closely related to a Schur complement type operator, corresponding to
the modified (regularized) bilinear form resulting from
$\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ by replacing
$\displaystyle c(\cdot,\cdot)$ with $\displaystyle(\cdot,\cdot)_{Q}$.

In order to present our main theoretical result, we give the following
definition.

######  Definition 1.

Two norms $\displaystyle\|\cdot\|_{Q}$ and $\displaystyle\|\cdot\|_{V}$ on the
Hilbert spaces $\displaystyle Q$ and $\displaystyle V$ are called fitted if
they satisfy the splittings (27) and (28), respectively, where
$\displaystyle|\cdot|_{Q}$ is a seminorm on $\displaystyle Q$ and
$\displaystyle|\cdot|_{V}$ and $\displaystyle|\cdot|_{b}$ are seminorms on
$\displaystyle V,$ the latter defined according to (29).

######  Theorem 5.

Let $\displaystyle\|\cdot\|_{V}$ and $\displaystyle\|\cdot\|_{Q}$ be fitted
norms according to Definition 1, which immediately implies the continuity of
$\displaystyle b(\cdot,\cdot)$ and $\displaystyle c(\cdot,\cdot)$ in these
norms with $\displaystyle\bar{C}_{b}=1$ and $\displaystyle\bar{C}_{c}=1$, cf.
(9)–(10). Consider the bilinear form
$\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ defined in (3) where
$\displaystyle a(\cdot,\cdot)$ is continuous, i.e., (8) holds, and
$\displaystyle a(\cdot,\cdot)$ and $\displaystyle c(\cdot,\cdot)$ are
symmetric positive semidefinite. Assume, further, that $\displaystyle
a(\cdot,\cdot)$ satisfies the coercivity estimate

(30) $a(v,v)\geq\underline{C}_{a}|v|_{V}^{2},\qquad\forall v\in V,$

and that there exists a constant $\displaystyle\underline{\beta}>0$ such that

(31) $\sup_{\begin{subarray}{c}v\in V\\\ v\neq
0\end{subarray}}\frac{b(v,q)}{\|v\|_{V}}\geq\underline{\beta}|q|_{Q},\qquad\forall
q\in Q.$

Then the bilinear form $\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$
is continuous and inf-sup stable under the combined norm
$\displaystyle\|\cdot\|_{Y}$ defined in (2), i.e., the conditions (16) and
(17) hold.

######  Remark 6.

The continuity of $\displaystyle b(\cdot,\cdot)$ readily follows from

$\displaystyle b(v,q)=\langle Bv,q\rangle_{Q^{\prime}\times
Q}=\langle\bar{Q}\bar{Q}^{-1}Bv,q\rangle_{Q^{\prime}\times
Q}=(\bar{Q}^{-1}Bv,q)_{Q}\leq\|\bar{Q}^{-1}Bv\|_{Q}\|q\|_{Q}\leq\|v\|_{V}\|q\|_{q}.$

If $\displaystyle|\cdot|_{V}$ is induced by the bilinear form $\displaystyle
a(\cdot,\cdot)$ then the continuity of $\displaystyle a(\cdot,\cdot)$ also
follows directly from the definition of the fitted norms.

######  Remark 7.

Theorem 5 is a generalization of Theorem 2 in the sense that given two norms
$\displaystyle\|\cdot\|_{Q_{B}}$ and $\displaystyle\|\cdot\|_{V_{B}}$ under
which the conditions of Theorem 2 are satisfied, one can always find two
fitted equivalent norms $\displaystyle\|\cdot\|_{Q}\eqsim\|\cdot\|_{Q_{B}}$
and $\displaystyle\|\cdot\|_{V}\eqsim\|\cdot\|_{V_{B}}$ such that the
conditions of Theorem 5 are satisfied in these fitted norms when
$\displaystyle c(\cdot,\cdot)\equiv 0$.

More specifically, for $\displaystyle c(\cdot,\cdot)\equiv 0$, we have
$\displaystyle|\cdot|_{Q}=||\cdot||_{Q}=\|q\|_{Q_{B}}$ and
$\displaystyle\bar{Q}=I$. If we define the fitted norm
$\displaystyle\|\cdot\|_{V}$ by choosing

(32) $\displaystyle\displaystyle|v|^{2}_{V}=a(v,v),$

then (30) obviously holds. In addition, there exits a constant
$\displaystyle\alpha_{0}$ such that (see [10, Proposition 4.3.4])

(33) $\displaystyle\displaystyle\alpha_{0}\|v\|^{2}_{V_{B}}\leq
a(v,v)+\|Bv\|^{2}_{Q^{\prime}}=\|v\|^{2}_{V}.$

At the same time, under the conditions of Theorem 2, the continuity of
$\displaystyle a(\cdot,\cdot)$ and $\displaystyle b(\cdot,\cdot)$ in the norm
$\displaystyle\|\cdot\|_{V_{B}}$ and $\displaystyle\|\cdot\|_{Q_{B}}$, we have

(34)
$\displaystyle\displaystyle\|v\|^{2}_{V}=a(v,v)+\|Bv\|^{2}_{Q^{\prime}}\leq
C\|v\|^{2}_{V_{B}}.$

Thus the fitted norm $\displaystyle\|\cdot\|_{V}$ is equivalent to the norm
$\displaystyle\|\cdot\|_{V_{B}}$, and (31) is induced by (20).

######  Remark 8.

Note that under the conditions of the theorem the coercivity of $\displaystyle
a(\cdot,\cdot)$ on $\displaystyle{\rm Ker}(B)$ in the (semi-) norms
$\displaystyle|\cdot|_{V}$ and $\displaystyle\|\cdot\|_{V}$ are equivalent
since $\displaystyle|v|_{V}=\|v\|_{V}$ for all $\displaystyle v\in{\rm
Ker}(B)$. The inf-sup condition (31), however, uses the seminorm
$\displaystyle|\cdot|_{Q}$ instead of $\displaystyle\|\cdot\|_{Q}$ as in
Brezzi's condition (20).

###### Proof of Theorem 5.

Demonstrating (16) is straightforward since

$\displaystyle\displaystyle\mathcal{A}((u;p),(v;q))=$
$\displaystyle\displaystyle a(u,v)+b(v,p)+b(u,q)-c(p,q)$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\bar{C}_{a}\|u\|_{V}\|v\|_{V}+\bar{C}_{b}(\|v\|_{V}\|p\|_{Q}+\|u\|_{V}\|q\|_{Q})+{\bar{C}_{c}}\|p\|_{Q}\|q\|_{Q}$
$\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle\bar{C}(\|u\|_{V}+\|p\|_{Q})(\|v\|_{V}+\|q\|_{Q})\leq
2\bar{C}\|(u;p)\|_{Y}\|(v;q)\|_{Y}$

with $\displaystyle\bar{C}:=\max\\{\bar{C}_{a},\bar{C}_{b},{\bar{C}_{c}}\\}$.

In order to prove (17) for a positive constant $\displaystyle\delta$, which
will be selected later, we choose

(36) $v:=\delta u+u_{0}$

where $\displaystyle u_{0}\in V$ is such that

(37a) $\displaystyle\displaystyle b(u_{0},p)=$
$\displaystyle\displaystyle|p|_{Q}^{2},$ (37b)
$\displaystyle\displaystyle\|u_{0}\|_{V}\leq$
$\displaystyle\displaystyle\underline{\beta}^{-1}|p|_{Q},$

and

(38) $q:=-\delta p+p_{0}$

where

(39) $p_{0}:=\bar{Q}^{-1}Bu.$

Note that the existence of an element $\displaystyle u_{0}$ satisfying (37)
follows from (31).

Then we have

$\displaystyle\displaystyle\|v\|_{V}\leq$ $\displaystyle\displaystyle\|\delta
u\|_{V}+\|u_{0}\|_{V}\leq\delta\|u\|_{V}+\underline{\beta}^{-1}|p|_{Q}\leq\delta\|u\|_{V}+\underline{\beta}^{-1}\|p\|_{Q},$
$\displaystyle\displaystyle\|q\|_{Q}\leq$
$\displaystyle\displaystyle\delta\|p\|_{Q}+\|p_{0}\|_{Q}=\delta\|p\|_{Q}+(\bar{Q}^{-1}Bu,\bar{Q}^{-1}Bu)_{Q}^{1/2}=\delta\|p\|_{Q}+|u|_{b},$

and, consequently,

$\displaystyle\displaystyle\|(v;q)\|^{2}_{Y}=$
$\displaystyle\displaystyle\|v\|_{V}^{2}+\|q\|_{Q}^{2}\leq
2(\delta^{2}+1)\|u\|_{V}^{2}+2(\underline{\beta}^{-2}+\delta^{2})\|p\|_{Q}^{2}.$

Hence, it follows

(42)
$\|(v;q)\|_{Y}\leq\left(2\max\\{(\delta^{2}+1),(\underline{\beta}^{-2}+\delta^{2})\\}\right)^{\frac{1}{2}}\|(u;p)\|_{Y}.$

Moreover, for the same choice of $\displaystyle v$ and $\displaystyle q$, we
obtain

$\displaystyle\displaystyle\allowdisplaybreaks\mathcal{A}((u;p),(v;q))$
$\displaystyle\displaystyle=a(u,\delta u+u_{0})+b(\delta u+u_{0},p)-b(u,\delta
p-p_{0})+c(p,\delta p-p_{0})$ $\displaystyle\displaystyle\geq\delta
a(u,u)+a(u,u_{0})+\delta b(u,p)+b(u_{0},p)-\delta b(u,p)+\delta c(p,p)$
$\displaystyle\displaystyle+{\langle
Bu,\bar{Q}^{-1}Bu\rangle_{Q^{\prime}\times Q}-\langle
Cp,\bar{Q}^{-1}Bu\rangle_{Q^{\prime}\times Q}}$
$\displaystyle\displaystyle\geq\delta
a(u,u)-\frac{1}{2}\epsilon^{-1}a(u,u)-\frac{1}{2}\epsilon
a(u_{0},u_{0})+|p|_{Q}^{2}+\delta c(p,p)$
$\displaystyle\displaystyle+{|u|^{2}_{b}-\frac{1}{2}\langle
Cp,\bar{Q}^{-1}Cp\rangle_{Q^{\prime}\times Q}-\frac{1}{2}\langle
Bu,\bar{Q}^{-1}Bu\rangle_{Q^{\prime}\times Q}}$
$\displaystyle\displaystyle\geq\left(\delta-\frac{1}{2}\epsilon^{-1}\right)a(u,u)-\frac{1}{2}\epsilon\bar{C}_{a}\underline{\beta}^{-2}|p|_{Q}^{2}+|p|_{Q}^{2}+\delta
c(p,p)$
$\displaystyle\displaystyle+{|u|^{2}_{b}-\frac{1}{2}\|p\|_{Q}^{2}-\frac{1}{2}|u|^{2}_{b}}$
$\displaystyle\displaystyle\geq\left(\delta-\frac{1}{2}\epsilon^{-1}\right)\underline{C}_{a}|u|_{V}^{2}+\left(1-\frac{1}{2}\epsilon\bar{C}_{a}\underline{\beta}^{-2}\right)|p|_{Q}^{2}+\delta
c(p,p){-\frac{1}{2}\|p\|_{Q}^{2}}+{\frac{1}{2}|u|_{b}^{2}}$

and, hence, for
$\displaystyle\epsilon=\frac{1}{2}\bar{C}_{a}^{-1}\underline{\beta}^{2}$ and
$\displaystyle\delta=\max\\{\frac{1}{4}\underline{C}_{a}^{-1}+\bar{C}_{a}\underline{\beta}^{-2},\frac{3}{4}\\}$,
we have

$\displaystyle\displaystyle\allowdisplaybreaks\mathcal{A}((u;p),(v;q))$
$\displaystyle\displaystyle\geq\left(\delta-{\bar{C}_{a}}\underline{\beta}^{-2}\right)\underline{C}_{a}|u|_{V}^{2}+{\frac{1}{4}}|p|_{Q}^{2}+{\left(\delta-\frac{1}{2}\right)}c(p,p)+{\frac{1}{2}|u|_{b}^{2}}$
(43)
$\displaystyle\displaystyle\geq{\frac{1}{4}}\left(\|u\|_{V}^{2}+\|p\|_{Q}^{2}\right)={\frac{1}{4}}\|(u;p)\|_{Y}^{2}.$

Together, (42) and (43) imply the inf-sup condition (17) which can
equivalently be formulated as

(44) $\sup_{(v;q)\in
Y}\frac{\mathcal{A}((u;p),(v;q))}{\|(v;q)\|_{Y}}\geq\underline{\alpha}\|(u;p)\|_{Y}\qquad\forall(u;p)\in
Y$

because the supremum on the left-hand side of (44) is bounded from below by

$\displaystyle\frac{\mathcal{A}((u;p),(v;q))}{\|(v;q)\|_{Y}}$

if we insert any fixed $\displaystyle(v;q)$, in particular the choice we made
and for which we proved

$\displaystyle\frac{\mathcal{A}((u;p),(v;q))}{\|(v;q)\|_{Y}}\geq\frac{\frac{1}{4}\|(u;p)\|_{Y}^{2}}{(2\max\\{(\delta^{2}+1),(\underline{\beta}^{-2}+\delta^{2})\\})^{1/2}\|(u;p)\|_{Y}}.$

∎

######  Remark 9.

The statement of Theorem 5 remains valid if the norms
$\displaystyle\|\cdot\|_{Q}$ and $\displaystyle\|\cdot\|_{V}$, as defined in
(27) and (28), are replaced with equivalent norms
$\displaystyle\|\cdot\|_{Q,c}\eqsim\|\cdot\|_{Q}$ and
$\displaystyle\|\cdot\|_{V,b}\eqsim\|\cdot\|_{V}$, hence using
$\displaystyle\|\cdot\|_{V,b}$ in (31) in this case. The proof remains
unchanged and the only difference in the final result is that the inf-sup
constant $\displaystyle\underline{\alpha}$ in (44) with respect to the (new)
equivalent combined norm has to be scaled by the quotient of the constants in
the norm equivalence relation for the combined norms. For that reason, without
loss of generality, we can use the fitted norms defined by (27) and (28)
directly in the formulation of the theorem.

##  3\. Application to poromechanics

In this section we consider several mixed variational problems arising in the
context of modeling flow in and deformation of porous media. All examples
except the first two are based on different formulations of Biot's
consolidation model. Here, we use bold letters to denote vector-valued
functions and the spaces to which they belong which means that we identify
certain non-bold symbols from the abstract framework in the previous section
with bold symbols, e.g., $\displaystyle v=\bm{v}$. To prove stability of these
formulations, we assume that proper boundary conditions are imposed and the
following classical inf-sup conditions, see [14], also [15, 10], for the pairs
of spaces $\displaystyle(\bm{V},Q)$ hold, i.e., there exist constants
$\displaystyle\beta_{d}$ and $\displaystyle\beta_{s}$ such that

(45) $\displaystyle\displaystyle\inf_{q\in Q}\sup_{\bm{v}\in\bm{V}}\frac{({\rm
div}\bm{v},q)}{\|\bm{v}\|_{\rm div}\|q\|}\geq\beta_{d}>0,$ (46)
$\displaystyle\displaystyle\inf_{q\in Q}\sup_{\bm{v}\in\bm{V}}\frac{({\rm
div}\bm{v},q)}{\|\bm{v}\|_{1}\|q\|}\geq\beta_{s}>0,$

where the norms $\displaystyle\|\cdot\|_{\rm div}$,
$\displaystyle\|\cdot\|_{1}$ and $\displaystyle\|\cdot\|$ denote the standard
$\displaystyle H({\rm div})$, $\displaystyle H^{1}$ and $\displaystyle L^{2}$
norms and $\displaystyle(\cdot,\cdot)$ is the $\displaystyle L^{2}$-inner
product.

######  Example 1.

The first example, taken from [11], is the mixed variational problem resulting
from a weak formulation of a generalized Poisson equation

$\displaystyle\displaystyle(\bm{u},\bm{v})+(p,{\rm div}\bm{v})$
$\displaystyle\displaystyle=0,\qquad\forall\bm{v}\in\bm{H}({\rm div},\Omega),$
$\displaystyle\displaystyle({\rm div}\bm{u},q)-t(p,q)$
$\displaystyle\displaystyle=-(f,q),\qquad\forall q\in L^{2}(\Omega),$

where $\displaystyle t\geq 0$ is a parameter. The bilinear forms generating
$\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ are given by

$\displaystyle a(\bm{u},\bm{v}):=(\bm{u},\bm{v}),\quad b(\bm{v},q):=({\rm
div}\bm{v},q),\quad c(p,q)=t(p,q),\quad\forall\bm{u},\bm{v}\in\bm{V},\forall
p,q\in Q,$

where $\displaystyle Q:=L^{2}(\Omega)$, $\displaystyle\bm{V}:=\bm{H}({\rm
div},\Omega)$. Using the norm fitting technique, we define
$\displaystyle|\cdot|_{Q}$, $\displaystyle|\cdot|_{V}$ by

$|q|_{Q}^{2}:=(q,q)\quad\forall q\in
Q\quad\text{and}\quad|\bm{v}|_{V}^{2}:=(\bm{v},\bm{v})\qquad\forall\bm{v}\in\bm{V}.$

Obviously, $\displaystyle B:={\rm div}:\bm{V}\rightarrow Q$, and (27) and (28)
take the form

$\|q\|_{Q}^{2}=|q|_{Q}^{2}+c(q,q)=(q,q)+t(q,q)=((1+t)q,q)=\langle(1+t)Iq,q\rangle_{Q^{\prime}\times
Q},$
$\|\bm{v}\|_{V}^{2}=|\bm{v}|_{V}^{2}+|\bm{v}|_{b}^{2}=(\bm{v},\bm{v})+\langle{\rm
div}\bm{v},\frac{1}{(1+t)}I{\rm div}\bm{v}\rangle_{Q^{\prime}\times
Q}=(\bm{v},\bm{v})+\frac{1}{(1+t)}({\rm div}\bm{v},{\rm div}\bm{v}),$

respectively. Since $\displaystyle
a(\bm{v},\bm{v})=(\bm{v},\bm{v})=|\bm{v}|_{V}^{2}$ for all
$\displaystyle\bm{v}\in\bm{V}$, condition (30) in Theorem 5 is satisfied with
$\displaystyle\underline{C}_{a}=1$.

Finally, we have to verify condition (31) in Theorem 5, namely

$\sup_{\bm{v}\in V}\frac{({\rm
div}\bm{v},q)}{(\|\bm{v}\|^{2}+\frac{1}{(1+t)}\|{\rm
div}\bm{v}\|^{2})^{1/2}}\geq\underline{\beta}|q|_{Q}=:\underline{\beta}\|q\|,\qquad\forall
q\in Q$

which follows directly from the classical inf-sup condition (45) on the spaces
$\displaystyle(\bm{V},Q)$ since $\displaystyle t\geq 0$.

######  Example 2.

The second example we consider is taken from [34]. Its mixed variational
formulation is given by

$\displaystyle\displaystyle(\nabla\bm{u},\nabla\bm{v})-(p,{\rm div}\bm{v})$
$\displaystyle\displaystyle=(\bm{f},\bm{v}),\qquad\forall\bm{v}\in\bm{H}_{0}^{1}(\Omega),$
$\displaystyle\displaystyle-({\rm div}\bm{u},q)-(\kappa\nabla p,\nabla q)$
$\displaystyle\displaystyle=(g,q),\qquad\forall q\in H^{1}(\Omega)\cap
L^{2}_{0}(\Omega),$

where $\displaystyle\kappa\geq 0$ is a parameter. The bilinear forms defining
$\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ here are given by

$\displaystyle a(\bm{u},\bm{v}):=(\nabla\bm{u},\nabla\bm{v}),\quad
b(\bm{v},q):=-({\rm div}\bm{v},q),\quad c(p,q)=(\kappa\nabla p,\nabla
q),\quad\forall\bm{u},\bm{v}\in\bm{V},\forall p,q\in Q.$

In this example, we set $\displaystyle Q:=H^{1}(\Omega)\cap
L^{2}_{0}(\Omega)$, $\displaystyle\bm{V}:=\bm{H}_{0}^{1}(\Omega)$, and
$\displaystyle|\cdot|_{Q}$, $\displaystyle|\cdot|_{V}$ to be

$|q|_{Q}^{2}:=(q,q)\quad\forall q\in
Q\quad\text{and}\quad|\bm{v}|_{V}^{2}:=(\nabla\bm{v},\nabla\bm{v})\qquad\forall\bm{v}\in\bm{V}.$

Then the operator $\displaystyle B$ is defined by $\displaystyle
B:\bm{V}\rightarrow Q$, $\displaystyle B:=-{\rm div}$ and the norm splittings
(27) and (28) are given by

(49) $\|q\|_{Q}^{2}=|q|_{Q}^{2}+c(q,q)=(q,q)+(\kappa\nabla q,\nabla
q)=\langle\bar{Q}q,q\rangle_{Q^{\prime}\times Q}$

and

$\|\bm{v}\|_{V}^{2}=|\bm{v}|_{V}^{2}+|\bm{v}|_{b}^{2}=(\nabla\bm{v},\nabla\bm{v})+\langle
B\bm{v},\bar{Q}^{-1}B\bm{v}\rangle_{Q^{\prime}\times Q}.$

Condition (30) is automatically satisfied with
$\displaystyle\underline{C}_{a}=1$. To show (31) we first note that from (49)
it follows that

$\displaystyle\langle\bar{Q}B\bm{v},B\bm{v}\rangle_{Q^{\prime}\times
Q}\geq(B\bm{v},B\bm{v}),$

or, equivalently,

$\displaystyle\langle B\bm{v},\bar{Q}^{-1}B\bm{v}\rangle_{Q^{\prime}\times
Q}\leq(B\bm{v},B\bm{v}).$

Thus

$\displaystyle\|\bm{v}\|_{V}^{2}=(\nabla\bm{v},\nabla\bm{v})+\langle
B\bm{v},\bar{Q}^{-1}B\bm{v}\rangle_{Q^{\prime}\times
Q}\leq(\nabla\bm{v},\nabla\bm{v})+({\rm div}\bm{v},{\rm div}\bm{v})\leq
2(\nabla\bm{v},\nabla\bm{v})\leq 2\|\bm{v}\|_{1}^{2}.$

Now, we choose $\displaystyle\bm{v}_{0}$ such that $\displaystyle-{\rm
div}\bm{v}_{0}=q$ and hereby obtain from the Stokes inf-sup condition (46) the
estimate $\displaystyle\|\bm{v}_{0}\|_{1}\leq\frac{1}{\beta_{s}}\|q\|$, and,
finally,

$\displaystyle\displaystyle\sup_{\bm{v}\in
V}\frac{b(\bm{v},q)}{\|\bm{v}\|_{V}}\geq\frac{b(\bm{v}_{0},q)}{\|\bm{v}_{0}\|_{V}}=\frac{\|q\|^{2}}{\|\bm{v}_{0}\|_{V}}\geq\frac{1}{\sqrt{2}}\frac{\|q\|^{2}}{\|\bm{v}_{0}\|_{1}}\geq\frac{\beta_{s}}{\sqrt{2}}\frac{\|q\|^{2}}{\|q\|}=:\underline{\beta}\|q\|=\underline{\beta}|q|_{Q},\qquad\forall
q\in Q.$

######  Example 3.

The two-field formulation of the quasi-static Biot's consolidation model after
semidiscretization in time by the implicit Euler method as studied in [34, 1],
reads

$\displaystyle\displaystyle(\varepsilon(\bm{u}),\varepsilon(\bm{v}))+\lambda({\rm
div}\bm{u},{\rm div}\bm{v})-\alpha(p_{F},{\rm div}\bm{v})$
$\displaystyle\displaystyle=(\bm{f},\bm{v}),\qquad\forall\bm{v}\in\bm{H}_{0}^{1}(\Omega),$
$\displaystyle\displaystyle-\alpha({\rm
div}\bm{u},q_{F})-{c_{0}}(p_{F},q_{F})-(\kappa\nabla p_{F},\nabla q_{F})$
$\displaystyle\displaystyle=(g,q_{F}),\qquad\forall q\in H_{0}^{1}(\Omega),$

where $\displaystyle\lambda\geq 0$ is a scaled Lamé coefficient,
$\displaystyle c_{0}\geq 0$ a storage coefficient, $\displaystyle\kappa$ the
(scaled) hydraulic conductivity, and $\displaystyle\alpha$ the (scaled) Biot-
Willis coefficient.

The bilinear forms defining
$\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ are given by

$\displaystyle\displaystyle a(\bm{u},\bm{v})$
$\displaystyle\displaystyle:=(\varepsilon(\bm{u}),\varepsilon(\bm{v}))+\lambda({\rm
div}\bm{u},{\rm div}\bm{v}),\qquad\forall\bm{u},\bm{v}\in\bm{V},$
$\displaystyle\displaystyle b(\bm{v},q_{F})$
$\displaystyle\displaystyle:=-\alpha({\rm
div}\bm{v},q_{F}),\qquad\forall\bm{v}\in\bm{V},\forall q_{F}\in Q,$
$\displaystyle\displaystyle c(p_{F},q_{F})$
$\displaystyle\displaystyle:={c_{0}}(p_{F},q_{F})+(\kappa\nabla p_{F},\nabla
q_{F}),\qquad\forall p_{F},q_{F}\in Q,$

where $\displaystyle Q:=H_{0}^{1}(\Omega)$,
$\displaystyle\bm{V}:=\bm{H}_{0}^{1}(\Omega)$. We define
$\displaystyle|\cdot|_{Q}$, $\displaystyle|\cdot|_{V}$ to be

$\displaystyle\displaystyle|q_{F}|_{Q}^{2}$
$\displaystyle\displaystyle:=\eta(q_{F},q_{F}),\qquad\forall q_{F}\in Q,$
$\displaystyle\displaystyle|\bm{v}|_{V}^{2}$
$\displaystyle\displaystyle:=(\varepsilon(\bm{v}),\varepsilon(\bm{v}))+\lambda({\rm
div}\bm{v},{\rm div}\bm{v}),\qquad\forall\bm{v}\in\bm{V},$

where the parameter $\displaystyle\eta>0$ is to be determined later. As
before, $\displaystyle a(\bm{v},\bm{v})\geq|\bm{v}|_{V}^{2}$ for all
$\displaystyle\bm{v}\in\bm{V}$, that is, (30) is satisfied with
$\displaystyle\underline{C}_{a}=1$. It remains to show (31). As in the
previous example, from the definition of $\displaystyle\|\cdot\|_{Q}$ it is
easy to conclude that

$\langle B\bm{v},\bar{Q}^{-1}B\bm{v}\rangle_{Q^{\prime}\times
Q}\leq\frac{1}{\eta}(B\bm{v},B\bm{v}),$

where $\displaystyle B:\bm{V}\rightarrow Q$, $\displaystyle B:=-\alpha{\rm
div}$. Therefore, we obtain

$\displaystyle\displaystyle\|\bm{v}\|_{V}^{2}$
$\displaystyle\displaystyle=(\varepsilon(\bm{v}),\varepsilon(\bm{v}))+\lambda({\rm
div}\bm{v},{\rm div}\bm{v})+\langle
B\bm{v},\bar{Q}^{-1}B\bm{v}\rangle_{Q^{\prime}\times Q}$
$\displaystyle\displaystyle\leq\left(1+\lambda+\frac{\alpha^{2}}{\eta}\right)(\nabla\bm{v},\nabla\bm{v})\leq\left(1+\lambda+\frac{\alpha^{2}}{\eta}\right)\|\bm{v}\|_{1}^{2}.$

We choose $\displaystyle\bm{v}_{0}$ such that $\displaystyle-{\rm
div}\bm{v}_{0}=\frac{1}{\sqrt{1+\lambda}}q_{F}$ and use (46) to obtain
$\displaystyle\|\bm{v}_{0}\|_{1}\leq\frac{1}{\beta_{s}}\frac{1}{\sqrt{1+\lambda}}\|q_{F}\|$,
and finally

(51) $\begin{array}[]{rl}\sup_{\bm{v}\in
V}\frac{b(\bm{v},q_{F})}{\|\bm{v}\|_{V}}&\geq\frac{b(\bm{v}_{0},q_{F})}{\|\bm{v}_{0}\|_{V}}=\frac{\frac{\alpha}{\sqrt{1+\lambda}}\|q_{F}\|^{2}}{\|\bm{v}_{0}\|_{V}}\geq\frac{\frac{\alpha}{\sqrt{1+\lambda}}}{\sqrt{\left(1+\lambda+\frac{\alpha^{2}}{\eta}\right)}}\frac{\|q_{F}\|^{2}}{\|\bm{v}_{0}\|_{1}}\\\
&\geq\frac{\beta_{s}\alpha}{\sqrt{\left(1+\lambda+\frac{\alpha^{2}}{\eta}\right)}}\frac{\|q_{F}\|^{2}}{\|q_{F}\|}=\frac{\beta_{s}\alpha}{\sqrt{\left(1+\lambda+\frac{\alpha^{2}}{\eta}\right)}}\frac{1}{\sqrt{\eta}}|q_{F}|_{Q}.\end{array}$

For $\displaystyle\eta:=\frac{\alpha^{2}}{(1+\lambda)}>0$ the right-hand side
of (51) is bounded from below by
$\displaystyle\frac{\beta_{s}}{\sqrt{2}}|q_{F}|_{Q}$ which shows (31) with
$\displaystyle\underline{\beta}=\frac{1}{\sqrt{2}}\beta_{s}$. Note that there
are also other possible choices for $\displaystyle\eta$.

######  Example 4.

By introducing $\displaystyle p_{S}=-\lambda{\rm div}\bm{u}$ and substituting
$\displaystyle\alpha p_{F}\rightarrow p_{F},c_{0}\alpha^{-2}\rightarrow
c_{0},\kappa\alpha^{-2}\rightarrow\kappa,\alpha^{-1}g\rightarrow g$ in Example
3 we obtain the following three-field variational formulation of Biot's model,
see [34],

$\displaystyle\displaystyle(\varepsilon(\bm{u}),\varepsilon(\bm{v}))-(p_{S}+p_{F},{\rm
div}\bm{v})$
$\displaystyle\displaystyle=(\bm{f},\bm{v}),\qquad\forall\bm{v}\in\bm{H}_{0}^{1}(\Omega),$
$\displaystyle\displaystyle-({\rm
div}\bm{u},q_{S})-\frac{1}{\lambda}(p_{S},q_{S})$
$\displaystyle\displaystyle=0,\qquad\forall q_{S}\in L_{0}^{2}(\Omega),$
$\displaystyle\displaystyle-({\rm
div}\bm{u},q_{F})-{c_{0}}(p_{F},q_{F})-(\kappa\nabla p_{F},\nabla q_{F})$
$\displaystyle\displaystyle=(g,q_{F}),\qquad\forall q_{F}\in
H_{0}^{1}(\Omega).$

The bilinear forms that determine
$\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ are

$\displaystyle\displaystyle a(\bm{u},\bm{v})$
$\displaystyle\displaystyle:=(\varepsilon(\bm{u}),\varepsilon(\bm{v})),\qquad\forall\bm{u},\bm{v}\in\bm{V},$
$\displaystyle\displaystyle b(\bm{v},\bm{q})$
$\displaystyle\displaystyle:=-({\rm div}\bm{v},q_{S})-({\rm
div}\bm{v},q_{F}),\qquad\forall\bm{v}\in\bm{V},\forall\bm{q}\in\bm{Q},$
$\displaystyle\displaystyle c(\bm{p},\bm{q})$
$\displaystyle\displaystyle:=\frac{1}{\lambda}(p_{S},q_{S})+{c_{0}}(p_{F},q_{F})+(\kappa\nabla
p_{F},\nabla q_{F}),\qquad\forall\bm{p},\bm{q}\in\bm{Q},$

where $\displaystyle\bm{V}=\bm{H}_{0}^{1}(\Omega)$,
$\displaystyle\bm{Q}=L_{0}^{2}(\Omega)\times H_{0}^{1}(\Omega)$ and
$\displaystyle\bm{p}=(p_{S};p_{F})$, $\displaystyle\bm{q}=(q_{S};q_{F})$. Then
the operator $\displaystyle B$ is given by

$\displaystyle B:=\begin{pmatrix}-{\rm div}\\\ -{\rm div}\end{pmatrix}.$

Moreover, we define $\displaystyle|\cdot|_{Q}$, $\displaystyle|\cdot|_{V}$ to
be

$\displaystyle\displaystyle|\bm{q}|_{Q}^{2}$
$\displaystyle\displaystyle:=\left(\begin{pmatrix}I&I\\\
I&I\end{pmatrix}\begin{pmatrix}q_{S}\\\
{q_{F,0}}\end{pmatrix},\begin{pmatrix}q_{S}\\\
{q_{F,0}}\end{pmatrix}\right)=\|q_{S}+{q_{F,0}}\|^{2},\qquad\forall\bm{q}\in\bm{Q},$
$\displaystyle\displaystyle|\bm{v}|_{V}^{2}$
$\displaystyle\displaystyle:=(\varepsilon(\bm{v}),\varepsilon(\bm{v})),\qquad\forall\bm{v}\in\bm{V},$

where $\displaystyle q_{F,0}$ is the $\displaystyle L^{2}$ projection of
$\displaystyle q_{F}$ to $\displaystyle L_{0}^{2}(\Omega)$. Then

$\displaystyle\displaystyle\|\bm{q}\|_{Q}$
$\displaystyle\displaystyle=\left(\begin{pmatrix}I&I\\\
I&I\end{pmatrix}\begin{pmatrix}q_{S}\\\
{q_{F,0}}\end{pmatrix},\begin{pmatrix}q_{S}\\\
{q_{F.0}}\end{pmatrix}\right)+\left(\begin{pmatrix}\frac{1}{\lambda}I&0\\\
0&{c_{0}}I-{\rm div}\kappa\nabla\end{pmatrix}\begin{pmatrix}q_{S}\\\
q_{F}\end{pmatrix},\begin{pmatrix}q_{S}\\\ q_{F}\end{pmatrix}\right)$
$\displaystyle\displaystyle=(\bar{Q}\bm{q},\bm{q})=\langle\bar{Q}\bm{q},\bm{q}\rangle_{\bm{Q}^{\prime}\times\bm{Q}}.$

As in the previous examples, (30) is satisfied with
$\displaystyle\underline{C}_{a}=1$. Next, we choose $\displaystyle\bm{v}_{0}$
such that $\displaystyle-{\rm div}\bm{v}_{0}=q_{S}+{q_{F,0}}$ for which we
have $\displaystyle\|\bm{v}_{0}\|_{1}\leq\beta_{s}^{-1}\|q_{S}+{q_{F,0}}\|$,
see (46). Then

$b(\bm{v}_{0},\bm{q})=\|q_{S}+{q_{F,0}}\|^{2}=|\bm{q}|_{Q}^{2}.$

Moreover,

$\displaystyle\displaystyle\|\bm{v}_{0}\|_{V}^{2}$
$\displaystyle\displaystyle=(\varepsilon(\bm{v}_{0}),\varepsilon(\bm{v}_{0}))+(\bar{Q}^{-1}B\bm{v}_{0},B\bm{v}_{0})=(\varepsilon(\bm{v}_{0}),\varepsilon(\bm{v}_{0}))+\left(\bar{Q}^{-1}\begin{pmatrix}q_{S}+{q_{F,0}}\\\
q_{S}+{q_{F,0}}\end{pmatrix},\begin{pmatrix}q_{S}+{q_{F,0}}\\\
q_{S}+{q_{F,0}}\end{pmatrix}\right)$
$\displaystyle\displaystyle\leq\|\bm{v}_{0}\|_{1}^{2}+\left(\bar{Q}^{-1}\begin{pmatrix}I&I\\\
I&I\end{pmatrix}\begin{pmatrix}q_{S}\\\
{q_{F,0}}\end{pmatrix},\begin{pmatrix}I&I\\\
I&I\end{pmatrix}\begin{pmatrix}q_{S}\\\ {q_{F,0}}\end{pmatrix}\right)$
$\displaystyle\displaystyle\leq\|\bm{v}_{0}\|_{1}^{2}+\left(\begin{pmatrix}I&I\\\
I&I\end{pmatrix}\begin{pmatrix}q_{S}\\\
{q_{F,0}}\end{pmatrix},\begin{pmatrix}q_{S}\\\ {q_{F,0}}\end{pmatrix}\right)$
$\displaystyle\displaystyle\leq\beta_{s}^{-2}\|q_{S}+{q_{F,0}}\|^{2}+\|q_{S}+{q_{F,0}}\|^{2}=(\beta_{s}^{-2}+1)|\bm{q}|_{Q}^{2}.$

Now (31) follows directly from

$\displaystyle\displaystyle\sup_{\bm{v}\in
V}\frac{b(\bm{v},\bm{q})}{\|\bm{v}\|_{V}}$
$\displaystyle\displaystyle\geq\frac{b(\bm{v}_{0},\bm{q})}{\|\bm{v}_{0}\|_{V}}\geq\frac{|\bm{q}|_{Q}^{2}}{\sqrt{(\beta_{s}^{-2}+1)}|\bm{q}|_{Q}}=:\underline{\beta}|\bm{q}|_{Q},\qquad\forall\bm{q}\in\bm{Q}.$

######  Remark 10.

In [34], the authors showed that the three-filed formulation for Biot's model
in Example 4 is not stable under the $\displaystyle Q$-seminorm defined by
$\displaystyle|\bm{q}|_{Q}^{2}=\|p_{S}\|^{2}+\|p_{F}\|^{2}$.

######  Example 5.

By introducing the total pressure $\displaystyle p_{T}=p_{S}+p_{F}$ in Example
4, another discrete in time three-field formulation of the quasi-static Biot's
consolidation model, see [34], is obtained and has the form

$\displaystyle\displaystyle(\varepsilon(\bm{u}),\varepsilon(\bm{v}))-(p_{T},{\rm
div}\bm{v})$
$\displaystyle\displaystyle=(\bm{f},\bm{v}),\qquad\forall\bm{v}\in\bm{H}_{0}^{1}(\Omega),$
$\displaystyle\displaystyle-({\rm
div}\bm{u},q_{T})-(\lambda^{-1}p_{T},q_{T})+(\alpha\lambda^{-1}p_{F},q_{T})$
$\displaystyle\displaystyle=0,\qquad\forall q_{T}\in L^{2}(\Omega),$
$\displaystyle\displaystyle(\alpha\lambda^{-1}p_{T},q_{F})-({(\alpha^{2}\lambda^{-1}+c_{0})}p_{F},q_{F})-(\kappa\nabla
p_{F},\nabla q_{F})$ $\displaystyle\displaystyle=(g,q_{F}),\qquad\forall q\in
H_{0}^{1}(\Omega).$

Here, $\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ is constructed
from

$\displaystyle\displaystyle a(\bm{u},\bm{v}):=$
$\displaystyle\displaystyle(\varepsilon(\bm{u}),\varepsilon(\bm{v})),\qquad\forall\bm{u},\bm{v}\in\bm{V},$
$\displaystyle\displaystyle b(\bm{v},\bm{q}):=$
$\displaystyle\displaystyle-({\rm
div}\bm{v},q_{T}),\qquad\forall\bm{v}\in\bm{V},\forall\bm{q}\in\bm{Q},$
$\displaystyle\displaystyle c(\bm{p},\bm{q}):=$
$\displaystyle\displaystyle(\lambda^{-1}p_{T},q_{T})-(\alpha\lambda^{-1}p_{F},q_{T})-(\alpha\lambda^{-1}p_{T},q_{F})+({(\alpha^{2}\lambda^{-1}+c_{0})}p_{F},q_{F})$
$\displaystyle\displaystyle+(\kappa\nabla p_{F},\nabla
q_{F}),\qquad\forall\bm{p},\bm{q}\in\bm{Q},$

where $\displaystyle\bm{V}=\bm{H}_{0}^{1}(\Omega)$,
$\displaystyle\bm{Q}=L^{2}(\Omega)\times H_{0}^{1}(\Omega)$ and
$\displaystyle\bm{p}=(p_{T};p_{F})$, $\displaystyle\bm{q}=(q_{T};q_{F})$.
Obviously, the operator $\displaystyle B:\bm{V}\rightarrow\bm{Q}$ is defined
by

$\displaystyle\displaystyle B:=\begin{pmatrix}-{\rm div}\\\ 0\end{pmatrix}.$

We next set

$\displaystyle\displaystyle|\bm{q}|_{Q}^{2}$
$\displaystyle\displaystyle:=(q_{T,0},q_{T,0}),\qquad\forall\bm{q}\in\bm{Q},$
$\displaystyle\displaystyle|\bm{v}|_{V}^{2}$
$\displaystyle\displaystyle:=(\varepsilon(\bm{v}),\varepsilon(\bm{v})),\qquad\forall\bm{v}\in\bm{V},$

where $\displaystyle q_{T,0}$ is the $\displaystyle L^{2}$ projection of
$\displaystyle q_{T}\in L^{2}(\Omega)$ to $\displaystyle L_{0}^{2}(\Omega)$.
Now we choose $\displaystyle\bm{v}_{0}\in\bm{V}=\bm{H}_{0}^{1}(\Omega)$ such
that $\displaystyle-{\rm div}\bm{v}_{0}=q_{T,0}$ for which it holds
$\displaystyle\|\bm{v}_{0}\|_{1}\leq\beta_{s}^{-1}\|q_{T,0}\|=\beta_{s}^{-1}|\bm{q}|_{Q}$,
see (46), and also

$\displaystyle\displaystyle
b(\bm{v}_{0},\bm{q})=(q_{T,0},q_{T,0})=|\bm{q}|_{Q}^{2}.$

From the definition of $\displaystyle\|\cdot\|_{Q}$, and using similar
arguments as in the previous examples, we obtain

$\displaystyle\displaystyle\|\bm{v}_{0}\|_{V}^{2}$
$\displaystyle\displaystyle=(\varepsilon(\bm{v}_{0}),\varepsilon(\bm{v}_{0}))+\langle
B\bm{v}_{0},\bar{Q}^{-1}B\bm{v}_{0}\rangle$
$\displaystyle\displaystyle\leq(\varepsilon(\bm{v}_{0}),\varepsilon(\bm{v}_{0}))+(B\bm{v}_{0},B\bm{v}_{0})\leq\|2\bm{v}_{0}\|_{1}^{2}\leq
2\beta_{s}^{-2}|\bm{q}|_{Q}^{2}.$

Again, (30) is satisfied with $\displaystyle\underline{C}_{a}=1$ while (31)
follows from

$\displaystyle\displaystyle\sup_{\bm{v}\in
V}\frac{b(\bm{v},\bm{q})}{\|\bm{v}\|_{V}}$
$\displaystyle\displaystyle\geq\frac{b(\bm{v}_{0},\bm{q})}{\|\bm{v}_{0}\|_{V}}\geq\beta_{s}\frac{|\bm{q}|_{Q}^{2}}{|\bm{q}|_{Q}}=:\underline{\beta}|\bm{q}|_{Q},\qquad\forall\bm{q}\in\bm{Q}.$

######  Remark 11.

Note that for the case $\displaystyle c_{0}=\alpha^{2}\lambda^{-1}$, as
considered in [34], we can further estimate

$\displaystyle\displaystyle c(\bm{q},\bm{q}):=$
$\displaystyle\displaystyle(\lambda^{-1}q_{T},q_{T})-2(\alpha\lambda^{-1}q_{F},q_{T})+2(\alpha^{2}\lambda^{-1}q_{F},q_{F})+(\kappa\nabla
q_{F},\nabla q_{F})$ $\displaystyle\displaystyle\geq$
$\displaystyle\displaystyle\frac{1}{4}(\lambda^{-1}q_{T},q_{T})+\frac{2}{3}(\alpha^{2}\lambda^{-1}q_{F},q_{F})+(\kappa\nabla
q_{F},\nabla q_{F})$ $\displaystyle\displaystyle\geq$
$\displaystyle\displaystyle\frac{1}{4}\left((\lambda^{-1}q_{T},q_{T})+(\alpha^{2}\lambda^{-1}q_{F},q_{F})+(\kappa\nabla
q_{F},\nabla q_{F})\right).$

Hence, we obtain

$\displaystyle\displaystyle\|\bm{q}\|^{2}_{Q}=|\bm{q}|^{2}_{Q}+c(\bm{q},\bm{q})\geq\frac{1}{4}\left((q_{T,0},q_{T,0})+(\lambda^{-1}q_{T},q_{T})+(\alpha^{2}\lambda^{-1}q_{F},q_{F})+(\kappa\nabla
q_{F},\nabla q_{F})\right),$

from which we conclude the stability result shown in [34]. The arguments
presented above are valid also for a vanishing storage coefficient, i.e.,
$\displaystyle c_{0}=0$. Moreover, our analysis shows how Example 4 and
Example 5 are related to each other. In fact, by the transformation
$\displaystyle\begin{pmatrix}p_{T}\\\ p_{F}\end{pmatrix}=\begin{pmatrix}I&I\\\
0&I\end{pmatrix}\begin{pmatrix}p_{S}\\\ p_{F}\end{pmatrix}$ or, equivalently,
$\displaystyle\begin{pmatrix}p_{S}\\\
p_{F}\end{pmatrix}=\begin{pmatrix}I&-I\\\
0&I\end{pmatrix}\begin{pmatrix}p_{T}\\\ p_{F}\end{pmatrix}$, we can derive the
stability of Example 4 and Example 5 from each other.

######  Example 6.

Next we consider a four-field formulation of Biot's model in which a total
pressure has been introduced [30]. The variational problem reads as

$\displaystyle\displaystyle
2\mu(\varepsilon(\bm{u}),\varepsilon(\bm{v}))+(p_{T},{\rm div}\bm{v})$
$\displaystyle\displaystyle=(\bm{f},\bm{v}),\qquad\forall\bm{v}\in\bm{H}_{0}^{1}(\Omega),$
$\displaystyle\displaystyle\frac{1}{\tau\kappa}(\bm{w},\bm{z})-(p,{\rm
div}\bm{z})$
$\displaystyle\displaystyle=0,\qquad\forall\bm{z}\in\bm{H}_{0}({\rm
div},\Omega),$ $\displaystyle\displaystyle({\rm
div}\bm{u},q_{T})-\frac{1}{\lambda}(p_{T},q_{T})-\frac{\alpha}{\lambda}(p,q_{T})$
$\displaystyle\displaystyle=0,\qquad\forall q_{T}\in L^{2}_{0}(\Omega),$
$\displaystyle\displaystyle-({\rm
div}\bm{w},q)-\frac{\alpha}{\lambda}(p_{T},q)-\left({c_{0}}+\frac{\alpha^{2}}{\lambda}\right)(p,q)$
$\displaystyle\displaystyle=(f,q),\qquad\forall q\in L^{2}_{0}(\Omega),$

with $\displaystyle\mu$ and $\displaystyle\lambda$ the Lamé coefficients,
$\displaystyle\tau$ the time step size, $\displaystyle\kappa$ the hydraulic
conductivity, $\displaystyle\alpha$ the Biot-Willis coefficient, and
$\displaystyle c_{0}\geq 0$ the constrained specific storage coefficient.

The bilinear forms defining
$\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ in this example are
given by

$\displaystyle\displaystyle a(\bar{\bm{u}},\bar{\bm{v}}):=$
$\displaystyle\displaystyle(\varepsilon(\bm{u}),\varepsilon(\bm{v}))+\frac{1}{\tau\kappa}(\bm{w},\bm{z}),\qquad\forall\bar{\bm{u}},\bar{\bm{v}}\in\bm{V},$
$\displaystyle\displaystyle b(\bar{\bm{v}},\bm{q}):=$
$\displaystyle\displaystyle({\rm div}\bm{v},q_{T})-({\rm
div}\bm{z},q),\qquad\forall\bar{\bm{v}}\in\bm{V},\forall\bm{q}\in\bm{Q},$
$\displaystyle\displaystyle c(\bm{p},\bm{q}):=$
$\displaystyle\displaystyle\frac{1}{\lambda}(p_{T},q_{T})+\frac{\alpha}{\lambda}(p,q_{T})+\frac{\alpha}{\lambda}(p_{T},q)+\left({c_{0}}+\frac{\alpha^{2}}{\lambda}\right)(p,q),\qquad\forall\bm{p},\bm{q}\in\bm{Q},$

where
$\displaystyle\bar{\bm{u}}:=(\bm{u};\bm{w}),\bar{\bm{v}}:=(\bm{v};\bm{z})\in\bm{V}=\bm{H}_{0}^{1}(\Omega)\times\bm{H}_{0}({\rm
div},\Omega),$
$\displaystyle\bm{p}:=(p_{T};p),\bm{q}:=(q_{T};q)\in\bm{Q}=L^{2}_{0}(\Omega)\times
L^{2}_{0}(\Omega).$ Hence, $\displaystyle B$ has the form

$\displaystyle\displaystyle B=\begin{pmatrix}{\rm div}&0\\\ 0&-{\rm
div}\end{pmatrix}.$

Using the norm fitting technique, we define

$\displaystyle\displaystyle|\bm{q}|_{Q}^{2}$
$\displaystyle\displaystyle:=\frac{1}{2\mu}(q_{T},q_{T})+\tau\kappa(q,q),\qquad\forall\bm{q}\in\bm{Q},$
$\displaystyle\displaystyle|\bar{\bm{v}}|_{V}^{2}$
$\displaystyle\displaystyle:=2\mu(\varepsilon(\bm{v}),\varepsilon(\bm{v}))+\frac{1}{\tau\kappa}(\bm{z},\bm{z}),\qquad\forall\bar{\bm{v}}\in\bm{V}.$

For this choice of $\displaystyle|\cdot|_{V}$ the coercivity estimate (30) is
again fulfilled with $\displaystyle\underline{C}_{a}=1$.

Now, in view of the Stokes and Darcy inf-sup conditions (46) and (45), for any
$\displaystyle\bm{q}=(q_{T};q)$ we can choose
$\displaystyle\bar{\bm{v}}_{0}=(\bm{v}_{0};\bm{z}_{0})$ such that
$\displaystyle{\rm div}\bm{v}_{0}=\frac{1}{2\mu}q_{T}$ and $\displaystyle-{\rm
div}\bm{z}_{0}=\tau\kappa\,q$ and it holds that

(55)
$\displaystyle\displaystyle\|\bm{v}_{0}\|_{1}\leq\frac{\beta_{s}^{-1}}{2\mu}\|q_{T}\|,\qquad\|\bm{z}_{0}\|_{\rm
div}\leq\beta_{d}^{-1}\tau\kappa\|q\|.$

Then we have

(56) $\displaystyle\displaystyle b(\bar{\bm{v}}_{0},\bm{q})=$
$\displaystyle\displaystyle({\rm div}\bm{v}_{0},q_{T})-({\rm
div}\bm{z}_{0},q)=\frac{1}{2\mu}(q_{T},q_{T})+\tau\kappa(q,q)=|\bm{q}|_{Q}^{2}$

and

$\displaystyle\displaystyle\|\bar{\bm{v}}_{0}\|_{V}^{2}$
$\displaystyle\displaystyle=2\mu(\varepsilon(\bm{v}_{0}),\varepsilon(\bm{v}_{0}))+\frac{1}{\tau\kappa}(\bm{z}_{0},\bm{z}_{0})+\frac{1}{\tau\kappa}({\rm
div}\bm{z}_{0},{\rm div}\bm{z}_{0})+\langle
B\bar{\bm{v}}_{0},\bar{Q}^{-1}B\bar{\bm{v}}_{0}\rangle$
$\displaystyle\displaystyle=2\mu(\varepsilon(\bm{v}_{0}),\varepsilon(\bm{v}_{0}))+\frac{1}{\tau\kappa}(\bm{z}_{0},\bm{z}_{0})+\frac{1}{\tau\kappa}({\rm
div}\bm{z}_{0},{\rm div}\bm{z}_{0})+\left(\bar{Q}^{-1}\begin{pmatrix}{\rm
div}\bm{v}_{0}\\\ -{\rm div}\bm{z}_{0}\end{pmatrix},\begin{pmatrix}{\rm
div}\bm{v}_{0}\\\ -{\rm div}\bm{z}_{0}\end{pmatrix}\right)$
$\displaystyle\displaystyle\leq
2\mu\|\bm{v}_{0}\|_{1}^{2}+\frac{1}{\tau\kappa}\|\bm{z}_{0}\|_{{\rm
div}}^{2}+\left(\begin{pmatrix}2\mu I&0\\\
0&\frac{1}{\tau\kappa}I\end{pmatrix}\begin{pmatrix}{\rm div}\bm{v}_{0}\\\
-{\rm div}\bm{z}_{0}\end{pmatrix},\begin{pmatrix}{\rm div}\bm{v}_{0}\\\ -{\rm
div}\bm{z}_{0}\end{pmatrix}\right)$
$\displaystyle\displaystyle=2\mu\|\bm{v}_{0}\|_{1}^{2}+\frac{1}{\tau\kappa}\|\bm{z}_{0}\|_{{\rm
div}}^{2}+2\mu\|{\rm div}\bm{v}_{0}\|^{2}+\frac{1}{\tau\kappa}\|{\rm
div}\bm{z}_{0}\|^{2}$ $\displaystyle\displaystyle\leq
4\mu\|\bm{v}_{0}\|_{1}^{2}+\frac{2}{\tau\kappa}\|\bm{z}_{0}\|_{{\rm
div}}^{2}.$

Now by (55) and the definition of $\displaystyle|\cdot|_{Q}$, we obtain

$\displaystyle\displaystyle\|\bar{\bm{v}}_{0}\|_{V}^{2}$
$\displaystyle\displaystyle\leq
4\mu\|\bm{v}_{0}\|_{1}^{2}+\frac{2}{\tau\kappa}\|\bm{z}_{0}\|_{{\rm
div}}^{2}\leq
4\mu\frac{\beta_{s}^{-2}}{4\mu^{2}}\|q_{T}\|^{2}+\frac{2}{\tau\kappa}\beta_{d}^{-2}\tau^{2}\kappa^{2}\|q\|^{2}$
(57) $\displaystyle\displaystyle\leq
2\max\\{\beta_{s}^{-2},\beta_{d}^{-2}\\}\left(\frac{1}{2\mu}\|q_{T}\|^{2}+\tau\kappa\|q\|^{2}\right)\leq
2\max\\{\beta_{s}^{-2},\beta_{d}^{-2}\\}|\bm{q}|^{2}_{Q}.$

Hence, in this example (31) follows from

$\displaystyle\displaystyle\sup_{\bar{\bm{v}}\in\bm{V}}\frac{b(\bar{\bm{v}},\bm{q})}{\|\bar{\bm{v}}\|_{V}}$
$\displaystyle\displaystyle\geq\frac{b(\bar{\bm{v}}_{0},\bm{q})}{\|\bar{\bm{v}}_{0}\|_{V}}\geq\frac{|\bm{q}|_{Q}^{2}}{\sqrt{2\max\\{\beta_{s}^{-2},\beta_{d}^{-2}\\}}|\bm{q}|_{Q}}=:\underline{\beta}|\bm{q}|_{Q},\qquad\forall\bm{q}\in\bm{Q},$

by combining (56) and (6).

######  Example 7.

Finally, let us consider the classical three-field formulation of Biot's
consolidation model as analyzed in [26]. After some rescaling of parameters
and semi-discretization in time by the implicit Euler method the static
variational problem (see equation (2.4) in [26]) to be solved in each time
step reads as

$\displaystyle\displaystyle(\varepsilon(\bm{u}),\varepsilon(\bm{v}))+\lambda_{\mu}({\rm
div}\bm{u},{\rm div}\bm{v})-(p,{\rm div}\bm{v})$
$\displaystyle\displaystyle=(\bm{f},\bm{v}),\qquad\forall\bm{v}\in\bm{H}_{0}^{1}(\Omega),$
$\displaystyle\displaystyle R_{p}^{-1}(\bm{w},\bm{z})-(p,{\rm div}\bm{z})$
$\displaystyle\displaystyle=0,\qquad\forall\bm{z}\in\bm{H}_{0}({\rm
div},\Omega)$ $\displaystyle\displaystyle-({\rm div}\bm{u},q)-({\rm
div}\bm{w},q)-\alpha_{p}(p,q)$ $\displaystyle\displaystyle=(g,q),\qquad\forall
q\in L^{2}_{0}(\Omega),$

with parameters $\displaystyle\lambda_{\mu}=\lambda/(2\mu)$, $\displaystyle
R_{p}^{-1}=\alpha^{2}\tau^{-1}\kappa^{-1}$, and
$\displaystyle\alpha_{p}=c_{0}\alpha^{-2}$. In this example, we set
$\displaystyle\bm{V}:=\bm{H}_{0}^{1}(\Omega)\times\bm{H}_{0}({\rm
div},\Omega)$, $\displaystyle Q:=L^{2}_{0}(\Omega)$. Obviously, the individual
bilinear forms that generate
$\displaystyle\mathcal{A}((\cdot;\cdot),(\cdot;\cdot))$ are defined by

$\displaystyle\displaystyle a(\bar{\bm{u}},\bar{\bm{v}})$
$\displaystyle\displaystyle=a((\bm{u};\bm{w}),(\bm{v};\bm{z}))$
$\displaystyle\displaystyle:=(\varepsilon(\bm{u}),\varepsilon(\bm{v}))+\lambda_{\mu}({\rm
div}\bm{u},{\rm
div}\bm{v})+R_{p}^{-1}(\bm{w},\bm{z}),\qquad\forall\bar{\bm{u}}=(\bm{u};\bm{w}),\bar{\bm{v}}=(\bm{v};\bm{z})\in\bm{V},$
$\displaystyle\displaystyle b(\bar{\bm{v}},q)$
$\displaystyle\displaystyle=b((\bm{v};\bm{z}),q)$
$\displaystyle\displaystyle:=-({\rm div}\bm{v}+{\rm
div}\bm{z},q)=:-(\text{Div}\bar{\bm{v}},q),\qquad\forall\bar{\bm{v}}=(\bm{v};\bm{z})\in\bm{V},\forall
q\in Q,$ $\displaystyle\displaystyle c(p,q)$
$\displaystyle\displaystyle:=\alpha_{p}(p,q),\qquad\forall p,q\in Q,$

where we have defined the operator $\displaystyle\text{Div}:\bm{V}\mapsto Q$
by $\displaystyle\text{Div}\bar{\bm{v}}=\text{Div}(\bm{v};\bm{z}):={\rm
div}\bm{v}+{\rm div}\bm{z}$ so that the operator $\displaystyle
B:\bm{V}\mapsto Q$ in this example is given by $\displaystyle B=-\text{Div}$.

The fitted norms we use in this example are determined by

$\displaystyle\displaystyle|q|_{Q}^{2}$
$\displaystyle\displaystyle:=\left(R_{p}+\frac{1}{1+\lambda_{\mu}}\right)(q,q),\qquad\forall
q\in Q,$ $\displaystyle\displaystyle|\bar{\bm{v}}|_{V}^{2}$
$\displaystyle\displaystyle:=a(\bar{\bm{v}},\bar{\bm{v}}),\qquad\forall\bar{\bm{v}}\in\bm{V}.$

The coercivity estimate (30) is again trivially fulfilled with
$\displaystyle\underline{C}_{a}=1$.

Next, from the inf-sup conditions (45) and (46) we infer that for any
$\displaystyle q\in Q$ there exist $\displaystyle\bm{v}_{0}$ and
$\displaystyle\bm{z}_{0}$ such that
$\displaystyle\bar{\bm{v}}_{0}:=(\bm{v}_{0};\bm{z}_{0})\in\bm{V}$ such that

(59a) $\displaystyle\displaystyle-{\rm div}\bm{z}_{0}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle R_{p}q\quad\mbox{
and }\quad\|\bm{z}_{0}\|^{2}_{\rm div}\leq\beta_{d}^{-2}R_{p}^{2}\|q\|^{2},$
(59b) $\displaystyle\displaystyle-{\rm div}\bm{v}_{0}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{1+\lambda_{\mu}}q\quad\mbox{ and
}\quad\|\bm{v}_{0}\|^{2}_{1}\leq\beta_{s}^{-2}\frac{1}{(1+\lambda_{\mu})^{2}}\|q\|^{2}.$

Now, from (59a) we conclude

(60) $\displaystyle\beta_{d}^{-2}R_{p}\|q\|^{2}$ $\displaystyle\geq
R_{p}^{-1}\|\bm{z}_{0}\|^{2}_{\rm div}=R_{p}^{-1}\left(\|{\rm
div}\bm{z}_{0}\|^{2}+\|\bm{z}_{0}\|^{2}\right)$
$\displaystyle=\|R_{p}^{-1/2}{\rm
div}\bm{z}_{0}\|^{2}+\|R_{p}^{-1/2}\bm{z}_{0}\|^{2},$

and from (59b)

(61) $\displaystyle\beta_{s}^{-2}\frac{1}{1+\lambda_{\mu}}\|q\|^{2}$
$\displaystyle\geq(1+\lambda_{\mu})\|\bm{v}_{0}\|^{2}_{1}=\frac{1}{2}(1+\lambda_{\mu})2\left(\|\nabla\bm{v}_{0}\|^{2}+\|\bm{v}_{0}\|^{2}\right)$
$\displaystyle\geq\frac{1}{2}\left[(\varepsilon(\bm{v}_{0}),\varepsilon(\bm{v}_{0}))+\lambda_{\mu}({\rm
div}\bm{v}_{0},{\rm div}\bm{v}_{0})+(1+\lambda_{\mu})({\rm div}\bm{v}_{0},{\rm
div}\bm{v}_{0})\right].$

Choosing
$\displaystyle\underline{\beta}:=\min\\{\beta_{d},\frac{\beta_{s}}{\sqrt{2}}\\}$,
we obtain

$\displaystyle\displaystyle\underline{\beta}^{-2}|q|_{Q}^{2}$
$\displaystyle\displaystyle=\underline{\beta}^{-2}\left(R_{p}+\frac{1}{1+\lambda_{\mu}}\right)\|q\|^{2}$
$\displaystyle\displaystyle\geq(\varepsilon(\bm{v}_{0}),\varepsilon(\bm{v}_{0}))+\lambda_{\mu}({\rm
div}\bm{v}_{0},{\rm div}\bm{v}_{0})+R_{p}^{-1}(\bm{z}_{0},\bm{z}_{0})$
$\displaystyle\displaystyle+R_{p}^{-1}({\rm div}\bm{z}_{0},{\rm
div}\bm{z}_{0})+\left(\frac{1}{1+\lambda_{\mu}}\right)^{-1}({\rm
div}\bm{v}_{0},{\rm div}\bm{v}_{0})$ (62)
$\displaystyle\displaystyle\geq|\bar{\bm{v}}_{0}|_{V}^{2}+\frac{1}{2}(R_{p}+\frac{1}{1+\lambda_{\mu}}+\alpha_{p})^{-1}(B\bar{\bm{v}}_{0},B\bar{\bm{v}}_{0}){\geq\frac{1}{2}\|\bar{\bm{v}}_{0}\|_{V}^{2}.}$

Hence, in this example (31) follows by combining (59a), (59b) and (7).

######  Remark 12.

Noting that for any $\displaystyle\epsilon\in(0,1)$, by Cauchy inequality, we
have

$\displaystyle\displaystyle\|\bar{\bm{v}}\|_{V}^{2}$
$\displaystyle\displaystyle=|\bar{\bm{v}}|_{V}^{2}+(\bar{Q}^{-1}B\bm{v},B\bm{v})=a(\bar{\bm{v}},\bar{\bm{v}})+(R_{p}+\frac{1}{1+\lambda_{\mu}}+\alpha_{p})^{-1}(B\bar{\bm{v}},B\bar{\bm{v}})$
$\displaystyle\displaystyle=(\varepsilon(\bm{v}),\varepsilon(\bm{v}))+\lambda_{\mu}({\rm
div}\bm{v},{\rm div}\bm{v})+R_{p}^{-1}(\bm{z},\bm{z})$
$\displaystyle\displaystyle+(R_{p}+\frac{1}{1+\lambda_{\mu}}+\alpha_{p})^{-1}({\rm
div}\bm{v}+{\rm div}\bm{z},{\rm div}\bm{v}+{\rm div}\bm{z})$
$\displaystyle\displaystyle\geq(\varepsilon(\bm{v}),\varepsilon(\bm{v}))+\lambda_{\mu}({\rm
div}\bm{v},{\rm div}\bm{v})+R_{p}^{-1}(\bm{z},\bm{z})$
$\displaystyle\displaystyle+(\frac{1}{1+\lambda_{\mu}})^{-1}(1-\epsilon^{-1})({\rm
div}\bm{v},{\rm
div}\bm{v})+(R_{p}+\frac{1}{1+\lambda_{\mu}}+\alpha_{p})^{-1}(1-\epsilon)({\rm
div}\bm{z},{\rm div}\bm{z})$
$\displaystyle\displaystyle\geq\frac{1}{2}(\varepsilon(\bm{v}),\varepsilon(\bm{v}))+\frac{1}{2}\lambda_{\mu}({\rm
div}\bm{v},{\rm div}\bm{v})+R_{p}^{-1}(\bm{z},\bm{z})$
$\displaystyle\displaystyle+({1+\lambda_{\mu}})(\frac{3}{2}-\epsilon^{-1})({\rm
div}\bm{v},{\rm
div}\bm{v})+(R_{p}+\frac{1}{1+\lambda_{\mu}}+\alpha_{p})^{-1}(1-\epsilon)({\rm
div}\bm{z},{\rm div}\bm{z}).$

Taking $\displaystyle\epsilon=\frac{2}{3}$, we have

$\displaystyle\displaystyle\|\bar{\bm{v}}\|_{V}^{2}$
$\displaystyle\displaystyle\geq\frac{1}{3}\left(\|\varepsilon(\bm{v})\|^{2}+\lambda_{\mu}\|{\rm
div}\bm{v}\|^{2}+R_{p}^{-1}\|\bm{z}\|^{2}+(R_{p}+\frac{1}{1+\lambda_{\mu}}+\alpha_{p})^{-1}\|{\rm
div}\bm{z}\|^{2}\right),$

from which we conclude the uniform stability result for classical three-field
formulation of Biot's consolidation model presented in [26].

##  4\. Conclusion

In this paper we have presented a new framework for the stability analysis of
perturbed saddle-point problems in variational formulation in a Hilbert space
setting. Our aproach is constructive and is based on a specific norm fitting
technique. The main theoretical result (Theorem 5) is a generalization of the
classical splitting theorem (Theorem 2) of Brezzi and allows to conclude the
necessary stability condition (big inf-sup condition) according to Babuška's
theory (Theorem 1) from conditions similar to those on which Brezzi's theorem
is based also in presence of a symmetric positive semidefinite perturbation
term $\displaystyle c(\cdot,\cdot)\not\equiv 0$.

As demonstrated on several examples, most of which were taken from different
formulations of Biot's consolidation model, the new norm fitting technique
guids the process of defining proper parameter-dependent norms and allows for
simple and short proofs of the stability of perturbed saddle-point problems.
Although the examples in the present paper are continuous (infinite-
dimensional) models, the abstract framework suggests that the proposed
technique would also be quite useful when studying the stability of various
mixed (finite element) discretizations and additionally apply to other
applications.

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