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 . Moreover, SPPs play a vital role in the context of image reconstruction , model order reduction , constrained optimization , optimal control , and parameter identification , 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 .
Saddle-point problems/systems are of a two-by-two block form and characterized by an operator/matrix of the form
where and denote positive semidefinite operators/matrices and the adjoint/transpose of an operator/matrix . We consider the symmetric case in this paper where , , and . Problems in which are often referred to as perturbed saddle-point problems. They are in the focus of this paper.
More general SPPs in which (and ) are allowed to be nonsymmetric and 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 rather than on , , , and separately.
Our work is motivated by the stability analysis of variational problems occurring in poromechanics (cf. ), 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 .
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 . 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 and equipped with the norms and induced by the scalar products and , respectively. We denote their product space by and endow it with the norm defined by
Next, we introduce an abstract bilinear form on defined by
for some symmetric positive semidefinite bilinear forms on , on , i.e.,
and a bilinear form on .
We assume that , and are continuous with respect to the norms and , i.e.,
Then each of these bilinear forms defines a bounded linear operator as follows:
Here and denote the dual spaces of and , respectively, and the corresponding duality pairing.
Associated with the bilinear form defined in (3) we consider the following abstract perturbed saddle-point problem
which can also be written as
thereby using the definitions and , or, in operator form
and , i.e., for all
The operator can also be represented in block form by
2.2. Babuška’s and Brezzi’s conditions for stability of saddle-point problems
Theorem 1 (Babuška ).
Let be a bounded linear functional. Then the saddle-point problem (12) is well-posed if and only if there exist positive constants and for which the conditions
hold. The solution then satisfies the stability estimate
For the classical saddle-point problem, i.e., , the following theorem which we formulate under the conditions that is symmetric positive semidefinite and
Note that if , the statement of the above theorem (Theorem 2) remains valid if we identify any two elements , for which is an element of , i.e., replacing the space with the quotient space and also the norm with , the latter being defined by
In this case, the solution is only unique up to an arbitrary element .
For the classical saddle-point problem Brezzi’s stability condition (20) and the continuity of imply Babuška’s stability condition (17), see , where it has also been shown that from (17) it follows (20) and the inf-sup condition for in the kernel of , the latter being equivalent to the coercivity estimate (19) if 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 , and in general are not sufficient to guarantee the stability condition (17) when . Additional conditions to ensure (17) have been studied, for example in [15, 12, 10].
In  it has been shown that a condition on the kernel of 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
and requires the following general assumption:
There exists a such that for every and for every it holds that the solution of (21) is bounded by .
The theorem then reads as follows:
Theorem 3 (Brezzi and Fortin ).
Assume that , and are continuous bilinear forms on , on , and on , respectively. Further assume that and are symmetric positive semidefinite. Finally, let (19), (20) (conditions (i) and (ii) from Theorem 2) and Assumption 1 be satisfied. Then for every and every problem (12) with as defined in (3) and has a unique solution in where
Moreover, the estimate
holds with a constant only depending on , , , and .
In order to ensure the boundedness (continuity) of the symmetric positive semidefinite bilinear form with respect to the norm , and more generally the boundedness of with respect to the combined norm , it is natural to include the contribution of in the norm , e.g., by defining via
for a proper seminorm or norm and a parameter .
As has been shown in  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
where is defined by
and provides a specific choice of , i.e., . The corresponding theorem then reads as:
Theorem 4 (Braess ).
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 and we choose in (24) for , then the perturbed saddle-point problem (12) is stable under the norm and the stability constant in (17) depends only on , and and the choice of .
Since (23) is an inf-sup condtion for which can be interpreted as a big inf-sup condition on for under the specific norm , see (26), Theorem 4 still does not provide us with the desired stability result in terms of conditions on , and separately. On the other hand, Theorem 3 requires the solution of the auxiliary problem (21) on 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 . 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 is not identical to zero. For fixed , the norm defined in (22) is equivalent to the norm defined by
where is a linear operator, which follows from the fact that the norm is induced by the scalar product on the Hilbert space . Now we introduce the following splitting of the norm defined by
where is a proper seminorm, which is a norm on satisfying
and is defined by
Here, is an isometric isomorphism since is an isometric isomorphism (Riesz isomorphism), i.e.,
Note that both and can be seminorms as long as they add up to a full norm. Likewise, only the sum of the seminorms and has to define a norm. In some particular situations, it is also useful to identify certain of the involved seminorms with , 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 by replacing with .
In order to present our main theoretical result, we give the following definition.
Let and be fitted norms according to Definition 1, which immediately implies the continuity of and in these norms with and , cf. (9)–(10). Consider the bilinear form defined in (3) where is continuous, i.e., (8) holds, and and are symmetric positive semidefinite. Assume, further, that satisfies the coercivity estimate
and that there exists a constant such that
The continuity of readily follows from
If is induced by the bilinear form then the continuity of also follows directly from the definition of the fitted norms.
Theorem 5 is a generalization of Theorem 2 in the sense that given two norms and under which the conditions of Theorem 2 are satisfied, one can always find two fitted equivalent norms and such that the conditions of Theorem 5 are satisfied in these fitted norms when .
More specifically, for , we have and . If we define the fitted norm by choosing
At the same time, under the conditions of Theorem 2, the continuity of and in the norm and , we have
Proof of Theorem 5.
Demonstrating (16) is straightforward since
In order to prove (17) for a positive constant , which will be selected later, we choose
where is such that
Then we have
Hence, it follows
Moreover, for the same choice of and , we obtain
and, hence, for and , we have
because the supremum on the left-hand side of (44) is bounded from below by
if we insert any fixed , in particular the choice we made and for which we proved
The statement of Theorem 5 remains valid if the norms and , as defined in (27) and (28), are replaced with equivalent norms and , hence using in (31) in this case. The proof remains unchanged and the only difference in the final result is that the inf-sup constant 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.,. To prove stability of these formulations, we assume that proper boundary conditions are imposed and the following classical inf-sup conditions, see , also [15, 10], for the pairs of spaces hold, i.e., there exist constants and such that
where the norms , and denote the standard , and norms and is the -inner product.
The first example, taken from , is the mixed variational problem resulting from a weak formulation of a generalized Poisson equation
where is a parameter. The bilinear forms generating are given by
where , . Using the norm fitting technique, we define , by