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

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)

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

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

More general SPPs in which $A$ (and $C$ ) are allowed to be nonsymmetric and $B_{1} \neq B_{2}$ 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 $A$ rather than on $A$ , $B_{1}$ , $B_{2}$ , and $C$ 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 $A$ .

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 $C \neq 0$ . 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 $V$ and $Q$ equipped with the norms $∥ \cdot ∥_{V}$ and $∥ \cdot ∥_{Q}$ induced by the scalar products $(\cdot, \cdot)_{V}$ and $(\cdot, \cdot)_{Q}$ , respectively. We denote their product space by $Y := V \times Q$ and endow it with the norm $∥ \cdot ∥_{Y}$ defined by

(2)

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

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

(3)

A ((u; p), (v; q)) := a (u, v) + b (v, p) + b (u, q) - c (p, q)

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

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

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

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

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

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


(11a)	$A : V \to V^{'} :$	$⟨ A u, v ⟩_{V^{'} \times V} = a (u, v), \forall u, v \in V,$
(11b)	$C : Q \to Q^{'} :$	$⟨ C p, q ⟩_{Q^{'} \times Q} = c (p, q), \forall p, q \in Q,$
(11c)	$B : V \to Q^{'} :$	$⟨ B v, q ⟩_{Q^{'} \times Q} = b (v, q), \forall v \in V, \forall q \in Q,$
(11d)	$B^{T} : Q \to V^{'} :$	$⟨ v, B^{T} q ⟩_{V \times V^{'}} = b (v, q), \forall v \in V, \forall q \in Q .$

Here $V^{'}$ and $Q^{'}$ denote the dual spaces of $V$ and $Q$ , respectively, and $⟨ \cdot, \cdot ⟩$ the corresponding duality pairing.

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

(12)

A ((u; p), (v; q)) = F ((v; q)) \forall v \in V, \forall q \in Q

which can also be written as

A (x, y) = F (y) \forall y \in Y,

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

(13)

A x = F

where

(14)

A : Y \to Y^{'} :

⟨ A x, y ⟩_{Y^{'} \times Y} = A (x, y), \forall x, y \in Y

and $F \in Y^{'}$ , i.e., $F : Y \to R : F (y) = ⟨ F, y ⟩_{Y^{'} \times Y}$ for all $y \in Y .$

The operator $A$ can also be represented in block form by

(15)

A = (\begin{matrix} A & B^{T} B & - C \end{matrix}) .

Problem (12) (and (13)) is called a perturbed saddle-point problem (in operator form) when $c (\cdot, \cdot) \equiv̸ 0$ and a classical saddle-point problem in the case $c (\cdot, \cdot) \equiv 0$ .

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

(16)

A (x, y) \leq ¯ C ∥ x ∥_{Y} ∥ y ∥_{Y} \forall x, y \in Y,

(17)

inf x \in Y sup y \in Y \frac{A (x, y)}{∥ x ∥_{Y} ∥ y ∥_{Y}} \geq α - - > 0

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

∥ x ∥_{Y} \leq \frac{1}{α - -} sup y \in Y \frac{F (y)}{∥ y ∥_{Y}} =: \frac{1}{α - -} ∥ F ∥_{Y^{'}} .

Remark 1.

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

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

(18)

K e r (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 $a (\cdot, \cdot)$ and $b (\cdot, \cdot)$ are continuous on $V \times V$ and on $V \times Q$ , respectively, $a (\cdot, \cdot)$ is symmetric positive semidefinite, and also that

(19)

a (v, v) \geq {C - -}_{a} ∥ v ∥_{V} \forall v \in K e r (B),

(20)

inf q \in Q sup v \in V \frac{b (v, q)}{∥ v ∥_{V} ∥ q ∥_{Q}} \geq β > 0,

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

Remark 2.

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

∥ q ∥_{Q / K e r (B^{T})} = inf q_{0} \in K e r (B^{T}) ∥ q + q_{0} ∥_{Q} .

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

For the classical saddle-point problem Brezzi’s stability condition (20) and the continuity of $a (\cdot, \cdot)$ 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 $a (\cdot, \cdot)$ in the kernel of $B$ , the latter being equivalent to the coercivity estimate (19) if $a (\cdot, \cdot)$ 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 $a (\cdot, \cdot)$ , $b (\cdot, \cdot)$ and $c (\cdot, \cdot)$ in general are not sufficient to guarantee the stability condition (17) when $c (\cdot, \cdot) \equiv̸ 0$ . 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 $B^{T}$ 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)

ϵ (p_{0}, q)_{Q} + c (p_{0}, q) = - c (p^{⊥}, q), \forall q \in K e r (B^{T})

and requires the following general assumption:

Assumption 1.

There exists a $γ_{0} > 0$ such that for every $p^{⊥} \in (K e r (B^{T}))^{⊥}$ and for every $ϵ > 0$ it holds that the solution $p_{0} \in K e r (B^{T})$ of (21) is bounded by $γ_{0} ∥ p_{0} ∥_{Q} \leq ∥ p^{⊥} ∥_{Q}$ .

The theorem then reads as follows:

Theorem 3 (Brezzi and Fortin [15]).

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

M = K e r (B^{T}) \cap K e r (C) .

Moreover, the estimate

∥ u ∥_{V} + ∥ p ∥_{Q / K e r (B^{T})} \leq K (∥ f ∥_{V^{'}} + ∥ g ∥_{Q^{'}})

holds with a constant $K$ only depending on ${¯ C}_{a}$ , ${¯ C}_{c}$ , ${C - -}_{a}$ , $β$ and $γ_{0}$ .

Remark 3.

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

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

(22)

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

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

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)

(24)

{| | | (v; q) | | |}^{2} := ∥ v ∥_{V}^{2} + | q |_{Q}^{2} + t^{2} c (q, q), t \in [0, 1],

and provides a specific choice of $∥ \cdot ∥_{Y}$ , i.e., . 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 $γ > 0$ and we choose $t > 0$ in (24) for $c (\cdot, \cdot) \equiv̸ 0$ , then the perturbed saddle-point problem (12) is stable under the norm and the stability constant $α - -$ in (17) depends only on $β$ , ${C - -}_{a}$ and $γ$ and the choice of $t$ .

Remark 4.

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

(25)

\frac{a (u, u)}{∥ u ∥_{V}} + sup q \in Q \frac{b (u, q)}{| q |_{Q} + t c (q, q)} \geq γ^{'} ∥ u ∥_{V} \forall u \in V .

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

(26)

sup (v; q) \in Y \frac{A ((u; 0), (v; q))}{| | | (v; q) | | |} \geq γ^{''} ∥ u ∥_{V} \forall u \in V .

Since (23) is an inf-sup condtion for $(a (\cdot, \cdot) + b (\cdot, \cdot))$ which can be interpreted as a big inf-sup condition on $A$ for $p = 0$ under the specific norm , see (26), Theorem 4 still does not provide us with the desired stability result in terms of conditions on $a (\cdot, \cdot)$ , $b (\cdot, \cdot)$ and $c (\cdot, \cdot)$ separately. On the other hand, Theorem 3 requires the solution of the auxiliary problem (21) on $K e r (B^{T})$ 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 $b (\cdot, \cdot)$ . 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 $c (\cdot, \cdot)$ is not identical to zero. For fixed $t > 0$ , the norm defined in (22) is equivalent to the norm defined by

(27)

∥ q ∥_{Q}^{2} := | q |_{Q}^{2} + c (q, q) =: ⟨ ¯ Q q, q ⟩_{Q^{'} \times Q}

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

(28)

∥ v ∥_{V}^{2}

:= | v |_{V}^{2} + | v |_{b}^{2}

where $| \cdot |_{V}$ is a proper seminorm, which is a norm on $K e r (B)$ satisfying

| v |_{V}^{2} ≂ a (v, v) \forall v \in K e r (B)

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

(29)

| v |_{b}^{2} := ⟨ B v, {¯ Q}^{- 1} B v ⟩_{Q^{'} \times Q} = ∥ B v ∥_{Q^{'}}^{2} .

Here, ${¯ Q}^{- 1} : Q^{'} \to Q$ is an isometric isomorphism since $¯ Q$ is an isometric isomorphism (Riesz isomorphism), i.e.,

∥ B v ∥_{Q^{'}}^{2}

= ∥ {¯ Q}^{- 1} B v ∥_{Q}^{2} = ({¯ Q}^{- 1} B v, {¯ Q}^{- 1} B v)_{Q} = ⟨ ¯ Q {¯ Q}^{- 1} B v, {¯ Q}^{- 1} B v ⟩_{Q^{'} \times Q} = ⟨ B v, {¯ Q}^{- 1} B v ⟩_{Q^{'} \times Q} .

Remark 5.

Note that both $| \cdot |_{V}$ and $| \cdot |_{b}$ can be seminorms as long as they add up to a full norm. Likewise, only the sum of the seminorms $| \cdot |_{Q}$ and $c (\cdot, \cdot)$ has to define a norm. In some particular situations, it is also useful to identify certain of the involved seminorms with $0$ , 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 $A ((\cdot; \cdot), (\cdot; \cdot))$ by replacing $c (\cdot, \cdot)$ with $(\cdot, \cdot)_{Q}$ .

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

Definition 1.

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

Theorem 5.

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

(30)

a (v, v) \geq {C - -}_{a} | v |_{V}^{2}, \forall v \in V,

and that there exists a constant $β - - > 0$ such that

(31)

sup \begin{matrix} v \in V v \neq 0 \end{matrix} \frac{b (v, q)}{∥ v ∥_{V}} \geq β - - | q |_{Q}, \forall q \in Q .

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

Remark 6.

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

b (v, q) = ⟨ B v, q ⟩_{Q^{'} \times Q} = ⟨ ¯ Q {¯ Q}^{- 1} B v, q ⟩_{Q^{'} \times Q} = ({¯ Q}^{- 1} B v, q)_{Q} \leq ∥ {¯ Q}^{- 1} B v ∥_{Q} ∥ q ∥_{Q} \leq ∥ v ∥_{V} ∥ q ∥_{q} .

If $| \cdot |_{V}$ is induced by the bilinear form $a (\cdot, \cdot)$ then the continuity of $a (\cdot, \cdot)$ 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 $∥ \cdot ∥_{Q_{B}}$ and $∥ \cdot ∥_{V_{B}}$ under which the conditions of Theorem 2 are satisfied, one can always find two fitted equivalent norms $∥ \cdot ∥_{Q} ≂ ∥ \cdot ∥_{Q_{B}}$ and $∥ \cdot ∥_{V} ≂ ∥ \cdot ∥_{V_{B}}$ such that the conditions of Theorem 5 are satisfied in these fitted norms when $c (\cdot, \cdot) \equiv 0$ .

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

(32)

| v |_{V}^{2} = a (v, v),

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

(33)

α_{0} ∥ v ∥_{V_{B}}^{2} \leq a (v, v) + ∥ B v ∥_{Q^{'}}^{2} = ∥ v ∥_{V}^{2} .

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

(34)

∥ v ∥_{V}^{2} = a (v, v) + ∥ B v ∥_{Q^{'}}^{2} \leq C ∥ v ∥_{V_{B}}^{2} .

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

Remark 8.

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

Proof of Theorem 5.

Demonstrating (16) is straightforward since


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

with $¯ C := max {{¯ C}_{a}, {¯ C}_{b}, {¯ C}_{c}}$ .

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

(36)

v := δ u + u_{0}

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


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

and

(38)

q := - δ p + p_{0}

where

(39)

p_{0} := {¯ Q}^{- 1} B u .

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

Then we have


	$∥ v ∥_{V} \leq$	$∥ δ u ∥_{V} + ∥ u_{0} ∥_{V} \leq δ ∥ u ∥_{V} + {β - -}^{- 1} \| p \|_{Q} \leq δ ∥ u ∥_{V} + {β - -}^{- 1} ∥ p ∥_{Q},$
	$∥ q ∥_{Q} \leq$	$δ ∥ p ∥_{Q} + ∥ p_{0} ∥_{Q} = δ ∥ p ∥_{Q} + ({¯ Q}^{- 1} B u, {¯ Q}^{- 1} B u)_{Q}^{1 / 2} = δ ∥ p ∥_{Q} + \| u \|_{b},$

and, consequently,


	$∥ (v; q) ∥_{Y}^{2} =$	$∥ v ∥_{V}^{2} + ∥ q ∥_{Q}^{2} \leq 2 (δ^{2} + 1) ∥ u ∥_{V}^{2} + 2 ({β - -}^{- 2} + δ^{2}) ∥ p ∥_{Q}^{2} .$

Hence, it follows

(42)

∥ (v; q) ∥_{Y} \leq {(2 max {(δ^{2} + 1), ({β - -}^{- 2} + δ^{2})})}^{\frac{1}{2}} ∥ (u; p) ∥_{Y} .

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

	$\allowdisplaybreaksA((u;p),(v;q))$	$= a (u, δ u + u_{0}) + b (δ u + u_{0}, p) - b (u, δ p - p_{0}) + c (p, δ p - p_{0})$
		$\geq δ a (u, u) + a (u, u_{0}) + δ b (u, p) + b (u_{0}, p) - δ b (u, p) + δ c (p, p)$
		$+ ⟨ B u, {¯ Q}^{- 1} B u ⟩_{Q^{'} \times Q} - ⟨ C p, {¯ Q}^{- 1} B u ⟩_{Q^{'} \times Q}$
		$\geq δ a (u, u) - \frac{1}{2} ϵ^{- 1} a (u, u) - \frac{1}{2} ϵ a (u_{0}, u_{0}) + \| p \|_{Q}^{2} + δ c (p, p)$
		$+ \| u \|_{b}^{2} - \frac{1}{2} ⟨ C p, {¯ Q}^{- 1} C p ⟩_{Q^{'} \times Q} - \frac{1}{2} ⟨ B u, {¯ Q}^{- 1} B u ⟩_{Q^{'} \times Q}$
		$\geq (δ - \frac{1}{2} ϵ^{- 1}) a (u, u) - \frac{1}{2} ϵ {¯ C}_{a} {β - -}^{- 2} \| p \|_{Q}^{2} + \| p \|_{Q}^{2} + δ c (p, p)$
		$+ \| u \|_{b}^{2} - \frac{1}{2} ∥ p ∥_{Q}^{2} - \frac{1}{2} \| u \|_{b}^{2}$

and, hence, for $ϵ = \frac{1}{2} {¯ C}_{a}^{- 1} {β - -}^{2}$ and $δ = max {\frac{1}{4} {C - -}_{a}^{- 1} + {¯ C}_{a} {β - -}^{- 2}, \frac{3}{4}}$ , we have

	$\allowdisplaybreaksA((u;p),(v;q))$	$\geq (δ - {¯ C}_{a} {β - -}^{- 2}) {C - -}_{a} \| u \|_{V}^{2} + \frac{1}{4} \| p \|_{Q}^{2} + (δ - \frac{1}{2}) c (p, p) + \frac{1}{2} \| u \|_{b}^{2}$
(43)			$\geq \frac{1}{4} (∥ u ∥_{V}^{2} + ∥ p ∥_{Q}^{2}) = \frac{1}{4} ∥ (u; p) ∥_{Y}^{2} .$

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

(44)

sup (v; q) \in Y \frac{A ((u; p), (v; q))}{∥ (v; q) ∥_{Y}} \geq α - - ∥ (u; p) ∥_{Y} \forall (u; p) \in Y

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

\frac{A ((u; p), (v; q))}{∥ (v; q) ∥_{Y}}

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

\frac{A ((u; p), (v; q))}{∥ (v; q) ∥_{Y}} \geq \frac{\frac{1}{4} ∥ (u; p) ∥_{Y}^{2}}{(2 max {(δ^{2} + 1), ({β - -}^{- 2} + δ^{2})})^{1 / 2} ∥ (u; p) ∥_{Y}} .

∎

Remark 9.

The statement of Theorem 5 remains valid if the norms $∥ \cdot ∥_{Q}$ and $∥ \cdot ∥_{V}$ , as defined in (27) and (28), are replaced with equivalent norms $∥ \cdot ∥_{Q, c} ≂ ∥ \cdot ∥_{Q}$ and $∥ \cdot ∥_{V, b} ≂ ∥ \cdot ∥_{V}$ , hence using $∥ \cdot ∥_{V, b}$ in (31) in this case. The proof remains unchanged and the only difference in the final result is that the inf-sup constant $α - -$ 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.,

v = 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

(V, Q)

hold, i.e., there exist constants

β_{d}

and

β_{s}

such that

(45)

inf q \in Q sup v \in V \frac{(d i v v, q)}{∥ v ∥_{d i v} ∥ q ∥} \geq β_{d} > 0,

(46)

inf q \in Q sup v \in V \frac{(d i v v, q)}{∥ v ∥_{1} ∥ q ∥} \geq β_{s} > 0,

where the norms $∥ \cdot ∥_{d i v}$ , $∥ \cdot ∥_{1}$ and $∥ \cdot ∥$ denote the standard $H (d i v)$ , $H^{1}$ and $L^{2}$ norms and $(\cdot, \cdot)$ is the $L^{2}$ -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


	$(u, v) + (p, d i v v)$	$= 0, \forall v \in H (d i v, Ω),$
	$(d i v u, q) - t (p, q)$	$= - (f, q), \forall q \in L^{2} (Ω),$

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

a (u, v) := (u, v), b (v, q) := (d i v v, q), c (p, q) = t (p, q), \forall u, v \in V, \forall p, q \in Q,

where $Q := L^{2} (Ω)$ , $V := H (d i v, Ω)$ . Using the norm fitting technique, we define $| \cdot |_{Q}$ , $| \cdot |_{V}$ by

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

Obviously, $B := d i v : V \to Q$ , 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) = ⟨ (1 + t) I q, q ⟩_{Q^{'} \times Q},

∥ v ∥_{V}^{2} = | v |_{V}^{2} + | v |_{b}^{2} = (v, v) + ⟨ d i v v, \frac{1}{(1 + t)} I d i v v ⟩_{Q^{'} \times Q} = (v, v) + \frac{1}{(1 + t)} (d i v v, d i v v),

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

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

sup v \in V \frac{(d i v v, q)}{(∥ v ∥^{2} + \frac{1}{(1 + t)} ∥ d i v v ∥^{2})^{1 / 2}} \geq β - - | q |_{Q} =: β - - ∥ q ∥, \forall q \in Q

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

Example 2.

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


	$(\nabla u, \nabla v) - (p, d i v v)$	$= (f, v), \forall v \in H_{0}^{1} (Ω),$
	$- (d i v u, q) - (κ \nabla p, \nabla q)$	$= (g, q), \forall q \in H^{1} (Ω) \cap L_{0}^{2} (Ω),$

where $κ \geq 0$ is a parameter. The bilinear forms defining $A ((\cdot; \cdot), (\cdot; \cdot))$ here are given by

a (u, v) := (\nabla u, \nabla v), b (v, q) := - (d i v v, q), c (p, q) = (κ \nabla p, \nabla q), \forall u, v \in V, \forall p, q \in Q .

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

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

Then the operator $B$ is defined by $B : V \to Q$ , $B := - d i v$ and the norm splittings (27) and (28) are given by

(49)

∥ q ∥

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

Authors

Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot's equations utilizing total pressure

Concentration of the Frobenius norms of pseudoinverses

Concentration of the Frobenius norms of generalized matrix inverses

Analysis of an abstract mixed formulation for viscoelastic problems

Understanding Adversarial Training: Increasing Local Stability of Neural Nets through Robust Optimization

A Framework for Automatic Monitoring of Norms that regulate Time Constrained Actions

Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations

1. Introduction

2. Abstract framework

2.1. Notation and problem formulation

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

Theorem 1 (Babuška [3]).

Remark 1.

Theorem 2.

Remark 2.

Assumption 1.

Theorem 3 (Brezzi and Fortin [15]).

Remark 3.

Theorem 4 (Braess [12]).

Remark 4.

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

Remark 5.

Definition 1.

Theorem 5.

Remark 6.

Remark 7.

Remark 8.

Proof of Theorem 5.

Remark 9.

3. Application to poromechanics

Example 1.

Example 2.

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

Authors

Related Research

Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot's equations utilizing total pressure

Concentration of the Frobenius norms of pseudoinverses

Concentration of the Frobenius norms of generalized matrix inverses

Analysis of an abstract mixed formulation for viscoelastic problems

Understanding Adversarial Training: Increasing Local Stability of Neural Nets through Robust Optimization

A Framework for Automatic Monitoring of Norms that regulate Time Constrained Actions

Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations

This week in AI

1. Introduction

2. Abstract framework

2.1. Notation and problem formulation

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

Theorem 1 (Babuška [3]).

Remark 1.

Theorem 2.

Remark 2.

Assumption 1.

Theorem 3 (Brezzi and Fortin [15]).

Remark 3.

Theorem 4 (Braess [12]).

Remark 4.

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

Remark 5.

Definition 1.

Theorem 5.

Remark 6.

Remark 7.

Remark 8.

Proof of Theorem 5.

Remark 9.

3. Application to poromechanics

Example 1.

Example 2.