Observations on degenerate saddle point problems
7 Observations on degenerate saddle point problems
Andrew V. Knyazev
Department of Mathematical Sciences
University of Colorado at Denver and Health Sciences Center
P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364
Abstract
We investigate degenerate saddle point problems, which can be viewed as limit cases of standard mixed formulations of symmetric
problems with large jumps in coefficients. We prove that they are well-posed in a standard norm despite the degeneracy. By
wellposedness we mean a stable dependence of the solution on the right-hand side. A known approach of splitting the saddle
point problem into separate equations for the primary unknown and for the Lagrange multiplier is used. We revisit the traditional
Ladygenskaya–Babuška–Brezzi (LBB) or inf–sup condition as well as the standard coercivity condition, and analyze how they
are affected by the degeneracy of the corresponding bilinear forms. We suggest and discuss generalized conditions that cover the
degenerate case. The LBB or inf–sup condition is necessary and sufficient for wellposedness of the problem with respect to the
Lagrange multiplier under some assumptions. The generalized coercivity condition is necessary and sufficient for wellposedness of
the problem with respect to the primary unknown under some other assumptions. We connect the generalized coercivity condition
to the positiveness of the minimum gap of relevant subspaces, and propose several equivalent expressions for the minimum gap. Our
results provide a foundation for research on uniform wellposedness of mixed formulations of symmetric problems with large jumps
in coefficients in a standard norm, independent of the jumps. Such problems appear, e.g., in numerical simulations of composite
materials made of components with contrasting properties.
Key words: Wellposedness, mixed, symmetric, saddle point, Lagrange multiplier, Ladygenskaya–Babuška–Brezzi (LBB) condition, inf–sup
condition, coercivity, minimum gap between subspaces.
PACS: 46.15.Cc, 02.30.Sa, 02.60.Lj
1. Introduction
Degenerate saddle point problems, e.g., can be viewed as
limit cases of mixed formulations of symmetric problems
with large jumps in coefficients, corresponding to an infi-
nite jump.We prove that the degeneracy does not affect the
wellposedness in a standard norm under some natural as-
sumptions, using ideas that are initiated by [3, 4, 5, 6, 7, 14,
15]. By wellposedness, contrary to illposedness, we mean a
stable dependence of the solution on the right-hand side.
Results of this paper provide a foundation for research on
uniform wellposedness of mixed formulations of symmet-
ric problems with large jumps in coefficients in a standard
norm, independent of the jumps.
Email address: andrew.knyazev[AT]cudenver.edu (Andrew V.
Knyazev).
URL: http://math.cudenver.edu/˜ aknyazev/ (Andrew V.
Knyazev).
1 Partially supported by the National Science Foundation award
DMS-0612751.
The necessary and sufficient condition, e.g., [9, 10], of
the standard wellposedness of an operator equation with
an arbitrary right–hand side is the existence of a bounded
inverse of the operator. We argue that in some practical
cases the equation is degenerate, i.e. the inverse operator
does not exist. Assuming that the right–hand side is in the
operator range, a solution exists, but is not unique. Tomake
the solution unique we factor out the operator null–space.
This leads to a natural generalization, where boundedness
of the pseudoinverse of the operator is used as the necessary
and sufficient condition of wellposedness of a degenerate
operator equation, by analogy with [13, 15].
With this idea in mind, we revisit necessary and sufficient
conditions of wellposedness of an abstract mixed problem.
In the symmetric case we consider here, the mixed problem
can be interpreted as a variational saddle point problem.
For generalized saddle point problems we refer the reader,
e.g., to [11].
We start in Section 2 with a standard abstract symmetric
mixed problem as in [9, 10]. By analogy with [14, 17], we
Preprint submitted to Comp. Meth. Applied Mech. Engineering, accepted and published as doi:10.1016/j.cma.2006.10.01926 October 2018
http://arxiv.org/abs/0704.1066v1
split the saddle point problem into two equations, for the
primary unknown and for the Lagrange multiplier. This
split is somewhat implicit in [9, 10]. The equation for the
primary unknown is self-consistent, since here we eliminate
the Lagrange multiplier from the mixed system using an
orthogonal projector.
Following, e.g., [10], we discuss the traditional neces-
sary and sufficient conditions of wellposedness, namely, the
Ladygenskaya–Babuška–Brezzi (LBB) or inf-sup condition
and the coercivity condition. The LBB or inf-sup condition,
considered in Section 3, is necessary and sufficient for a sta-
ble dependence of the Lagrange multiplier on an arbitrary
right-hand side.
We review the traditional point of view that the coerciv-
ity condition is a necessary and sufficient condition of well-
posedness of the problem. In Section 4, an operator form
of the dual variational problem without assuming the co-
ercivity condition is considered. We examine the unique-
ness of the solution and describe all possible multiple solu-
tions for a given right-hand side. All admissible right-hand
sides are determined. We formulate several equivalent nec-
essary and sufficient conditions of wellposedness in terms
of closedness of relevant subspaces. We also derive a geo-
metrical condition—a positiveness of a minimum gap [12]
between relevant closed subspaces.
A possible application of our theory is the Hellinger–
Reissner formulation, e.g., [1], of nonhomogeneous Lamé
equations for media with (almost) rigid inclusions, where
the Lagrange multiplier is the displacement, and we get an
operator equation for the stress on the closed subspace of
divergence free (in a weak sense) stresses. Infinitely large
Lamé coefficients λ and µ, in a subdomain, result in a
null-space of the operator in the equation for the stress,
so the inverse operator does not exist and the problem is
not wellposed in a traditional sense. Our abstract geomet-
rical condition of generalized wellposedness in this example
is equivalent to a possibility of extension of displacements
preserving the energy norm of the Lamé operator. It has
been proved in [4, 7] that such an extension is possible un-
der some assumptions. We expect that in the limit case of
infinitely large Lamé coefficients λ and µ in a subdomain
the pseudoinverse of the operator is bounded, which makes
the problem wellposed for the stresses in the L2 sense, i.e.
the L2 norm of the stress is stable even if the Lamé coef-
ficients are large in a subdomain. We plan to address this
application in the future.
2. Abstract symmetric saddle point problems
In this section we essentially follow well known argu-
ments, e.g., [10], with some simplifications due to the sym-
metry of the saddle point problem and our unwillingness
to introduce dual spaces. Straightforward manipulations,
using a pair of complementary closed subspaces, allow us,
as in [14, 17], to formulate separate equations for the pri-
mary unknown and for the Lagrange multiplier of the sad-
dle point problem; see, e.g., survey [8, Sec. 6] for similar
matrix null-space methods. We start by formulating and
investigating the problem using bilinear forms, and then
repeat the arguments for operator-based formulations that
are used in the last section of the paper.
2.1. Formulations using bilinear forms
LetH andV be two real Hilbert spaces with scalar prod-
ucts and norms denoted by (·, ·)H, ‖ · ‖H and (·, ·)V, ‖ · ‖V
correspondingly. Let a(·, ·) : H × H → R and b(·, ·) :
H ×V → R be two continuous bilinear forms with a(·, ·)
symmetric and nonnegative definite. We consider the fol-
lowing problem: for a given g ∈ H and f ∈ H find σ ∈ H,
called the “primary unknown,” and u ∈ V, called the “La-
grange multiplier,” such that
a(σ, ǫ) + b(ǫ, u) = (g, ǫ)H, ∀ǫ ∈ H,
b(σ − f, v) = 0, ∀v ∈ V.
We place the right-hand side f “inside” of the form b as it
allows us to take f ∈ H, not to introduce the dual space
V′, and makes several statements somewhat simpler. We
call (1) a saddle point problem, since equations (1) are the
optimality conditions and their solution is a saddle point
for the Lagrangian, e.g., [10], defined by a(σ, σ) + 2b(σ −
f, u)− 2(g, σ)H.
We call a linear manifold, not necessarily closed, a “sub-
space” and a closed linear manifold a “closed subspace.”
Let us introduce a special notation N ⊆ H for the closed
subspace, which is the null-space of the bilinear form b(·, ·)
with respect to its first argument, i.e. N = {ǫ ∈ H :
b(ǫ, v) = 0, ∀v ∈ V}. Let us denote by P ≡ N⊥ ⊆ H the
closed subspace which is H-orthogonal (complementary)
to N. Closed subspaces N and P play important roles in
this paper, so let us introduce anH-orthogonal projector P
on H such that N(P ) = N and R(P ) = P and the com-
plementary projector P⊥ = I − P with R(P⊥) = N and
N(P⊥) = P, where by R(P ) we denote the range of opera-
tor P and, with a slight abuse of the notation, by N(P ) we
denote the null-space of operator P . We assume through-
out the paper, unless stated otherwise, that a bounded op-
erator is defined everywhere on a corresponding space. As
an orthogonal projector, operator P : H → H is bounded
H-selfadjoint, P = P ∗, and satisfies P = P 2.
In the first equation of system (1), let us split it into two
equations, by plugging separately ǫ = Pǫ ∈ P and ǫ =
P⊥ǫ ∈ N and using the fact that b(P⊥ǫ, u) = 0, ∀ǫ ∈ H.
The second equation in system (1) has a simple equivalent
geometric interpretation: σ − f ∈ N, or (σ − f, ǫ)H =
0, ∀ǫ ∈ P. We then rewrite system (1) in the following
equivalent form:
a(σ, ǫ) + b(ǫ, u) = (g, ǫ)H, ∀ǫ ∈ P,
a(σ, ǫ) = (g, ǫ)H, ∀ǫ ∈ N,
(σ − f, ǫ)H = 0, ∀ǫ ∈ P.
Nowwemake an important observation that we can treat
the first line in system (2) as an equation for the Lagrange
multiplier u, given the primary unknown σ, i.e.
b(ǫ, u) = (g, ǫ)H − a(σ, ǫ), ∀ǫ ∈ P. (3)
The last two lines in system (2) involve neither the Lagrange
multiplier u, nor the bilinear form b, and can be used to
determine the primary unknown σ:
a(σ, ǫ) = (g, ǫ)H, ∀ǫ ∈ N,
(σ − f, ǫ)H = 0, ∀ǫ ∈ P.
System (4) describes, e.g., [10], the optimality conditions
of the constrained minimization problem inf {a(σ, σ) −
2(g, σ)H}, σ ∈ H : (σ − f, ǫ)H = 0, ∀ǫ ∈ P.
2.2. Operator-based formulations
In addition to the formulations above involving bilin-
ear forms, it is convenient to consider equivalent operator-
based formulations.We associatewith the forms a and b two
linear continuous operators A : H → H and B : H → V
defined by (Aσ, ǫ)H = a(σ, ǫ), (Bσ, v)V = b(σ, v), ∀ǫ, σ ∈
H, v ∈ V. In this definition of A and B we follow a
slightly simplified, e.g., [11, 17], rather than standard [10],
approach, namely, we do not need dual spaces H′ and V′.
Now, we reformulate the main statements of subsection 2.1
using the just defined operators A and B. The following
operator formulation
Aσ + B∗u = g in H,
B(σ − f) = 0 in V
is equivalent to the original problem (1) with the bilinear
forms, where the adjoint operator B∗ : V → H is defined,
as usual, by (σ,B∗v)H = (Bσ, v)V, ∀σ ∈ H, v ∈ V. The
operator A is selfadjoint and nonnegative definite, A =
A∗ ≥ 0 on H since it is defined by the symmetric and
nonnegative definite form a.
We notice that the second equation in system (5) has
the same geometric interpretation as in the case of bilinear
forms-based system (1): σ − f ∈ N(B). The null-space
N(B) ⊆ H and its H-orthogonal complement R(B∗) ⊆ H
have already been denoted by N and P, correspondingly,
and introduced together with the H-orthogonal projector
P on H such that N = N(P ) = N(B) and P = R(P ) =
R(B∗) in the previous subsection.
We split the first equation in system (5) in two orthogonal
parts corresponding to N and P, using that PB∗u = B∗u
and P⊥B∗u = 0, since R(B∗) ⊆ P. We replace B with P ,
since they share the same null-space, in the second equation
in system (5) to get the following equivalent form of system
PAσ +B∗u = Pg in H,
P⊥Aσ = P⊥g in H,
P (σ − f) = 0 in H.
We notice that the first line in system (6) is an equation
for the Lagrange multiplier u, given the primary unknown
σ, as in (3), i.e. B∗u = P (g −Aσ).
We next discuss the necessary and sufficient conditions
from [10] of wellposedness of the problem and make it clear
why one can find weaker necessary and sufficient conditions.
To simplify our arguments, we take advantage in the rest
of the paper of the split of the original system into separate
equations for the Lagrange multiplier u and the primary
unknown σ that we have described in this section. It is
important to realize, however, that we have not made any
substitutions, neither in the solutions u and σ, nor in the
right-hand sides f and g. So whatever statements we next
prove concerning the dependence of the solutions u and
σ on the right-hand sides f and g, these statements are
equally applicable to both the separate equations and to the
original system in either bilinear form- or operator-based
context.
3. Inf-sup or LBB condition
In this section, we discuss a traditional assumption, be-
ing recently referred to as Ladygenskaya–Babuška–Brezzi
(LBB) condition, see Babuška and Aziz [2], Brezzi and
Fortin [10], Ladyzhenskaya [16], that the range of operator
B : H → V, denoted by R(B), is closed. The closedness of
a range of a closed operator is ultimately connected to the
boundedness of the operator (pseudo-)inverse, e.g., [12].
In our specific situation, operator B is bounded with
the closed domain H and, thus, is closed, so its (pseudo-
)inverse B−1 : R(B) → H/N(B) is also closed. It is neces-
sary to use a factor-space here to define the inverse, since
the standard operator inverse B−1 : R(B) → H does not
exist if N(B) is nontrivial. We note that N(B) is closed
and that the factor-space H/N(B) is a Hilbert space, as
is H. In a Hilbert space, a convenient set of representants
for the classes in the factor-space is simply the correspond-
ing orthogonal complement, e.g., H/N(B) is isometrically
isomorphic to P = (N(B))⊥ ⊆ H, so we set ‖σ‖H/N(B) =
‖Pσ‖H. The subspaceR(B) is the domain of the closed op-
erator B−1 : R(B) → H/N(B) therefore, R(B) is closed
if and only if B−1 : R(B) → H/N(B) is bounded. Closed-
ness of R(B) is equivalent to closedness of R(B∗), so all
the arguments above can be equivalently reformulated for
the adjoint operator B∗ and its (pseudo-)inverse.
When written in terms of inequalities involving the bi-
linear form b :
b(σ, v)
‖σ‖H/N(B)‖v‖V
= inf
‖Bσ‖V
‖σ‖H/N(B)
‖B−1‖R(B)→H/N(B)
or, equivalently,
b(σ, v)
‖σ‖H‖v‖V/N(B∗)
= inf
‖B∗v‖H
‖v‖V/N(B∗)
‖B−∗‖R(B∗)→V/N(B∗)
the LBB condition is also known as the inf-sup condition,
see Babuška and Aziz [2], Brezzi and Fortin [10], where
V/N(B∗) means the factor-space of V with respect to the
closed subspace N(B∗). We implicitly assume that the ar-
guments in the inf-sup formulas above and throughout the
paper do not make both the numerator and the denomi-
nator vanish. In Ladyzhenskaya [16], the inf-sup condition
does not appear to be explicitly formulated, instead, closed-
ness of a range of the gradient operator is investigated in
connection with wellposedness of the diffusion equation.
We note that the induced norms of an operator and its
adjoint are equal, so both inf-sup expressions above are
equal to the same constant that we call cb. If at least one
of the spaces H or V is finite dimensional then the value cb
is positive automatically, so it becomes important how cb
depends on some parameters, e.g., on the dimension.
Let us mention that in many practical applications the
space V can be naturally defined such that N(B∗) = {0},
so the latter inf-sup expression of the LBB condition takes
the form
b(σ, v)
‖σ‖H‖v‖V
= cb > 0,
which can be most often seen in publications on the subject.
We now contribute our own equivalent formulations of the
LBB condition.
Lemma 3.1 Subspaces R(B) ⊆ V and R(BB∗) ⊆ V are
closed simultaneously. Moreover, if either of them is closed
we have R(BB∗) = R(B).
Proof. If BB∗v = 0 then (B∗v,B∗v)H = 0, i.e. B
∗v = 0,
which proves that N(BB∗) = N(B∗). Taking an orthogo-
nal complement to both parts givesR(BB∗) = R(B) as the
operator BB∗ is selfadjoint. Trivially, R(BB∗) ⊆ R(B).
If the range R(BB∗) is closed then R(B) = R(BB∗) =
R(BB∗) ⊆ R(B), but clearlyR(B) ⊆ R(B), which proves
closedness of R(B) = R(BB∗).
To prove the inverse statement, assuming that R(B)
is closed, we invoke the orthogonal decomposition argu-
ment 2 : H = R(B∗) ⊕ (R(B∗))⊥ = R(B∗) ⊕ N(B) since
R(B) and thus R(B∗) are closed. Multiplying this equal-
ity by B gives R(B) = BH = B(R(B∗) ⊕ N(B)) =
BR(B∗) = R(BB∗). ✷
2 This proof is suggested by an anonymous referee
We use the previous lemma to introduce (BB∗)−1 :
R(BB∗) → V/N(B∗) in the next Lemma 3.2. It is nec-
essary to use the factor-space V/N(B∗) here, since the
standard inverse (BB∗)−1 : R(BB∗) → V does not exist
if N(B∗) is nontrivial.
Lemma 3.2 Closedness of R(B) ⊆ V is equivalent
to boundedness of the operator (BB∗)−1 : R(BB∗) →
V/N(B∗).
Proof. By Lemma 3.1, closedness of R(B) ⊆ V is equiv-
alent to closedness of R(BB∗) ⊆ V. We use several well-
known statements on closed operators, e.g., [12], applied
to the operator BB∗, that we have already reviewed in the
second paragraph of this section for the operator B. The
operator BB∗ is bounded and has the closed domain V, so
the operator is closed and its (pseudo-)inverse (BB∗)−1 :
R(BB∗) → V/N(B∗) with the domain R(BB∗) ⊆ V is
also closed. The domain R(BB∗) ⊆ V of the closed opera-
tor B−1 : R(BB∗) → H/N(B) is closed if and only if the
operator is bounded. ✷
If R(B) is closed then, using Lemmas 3.1 and 3.2,
R(B) = R(BB∗) and we can derive the following useful
formula
P = B∗(BB∗)−1B : H → H. (7)
Indeed, we first note that R((BB∗)−1) ⊆ V/N(B∗) is
multiplied by B∗ in (7), so the product is independent
of the choice of a representant from the equivalence class
V/N(B∗) and, thus, is correctly defined. Second, righ-hand
side of (7) is a linear and bounded operator as a product of
linear and bounded operators. Moreover, it is an orthogonal
projector on H since it is selfadjoint and idempotent, and
has the null-space the same as the orthoprojector P has.
If the LBB condition is not satisfied, i.e. R(B) is not
closed, then the domain of definition of the operator
B∗(BB∗)−1B is the subspace R(B∗) ⊕ N(B), which is
not closed, and formula (7), where P is the orthogonal
projector on H with N(P ) = N(B), clearly does not hold.
Let us note that in the case of finite dimensional spaces
H and V the range R(B) is evidently closed, the opera-
tor (B∗)+ = (BB∗)+B is the well-known Moore–Penrose
pseudo inverse of B∗, and P = B∗(B∗)+ is the well known
formula for the orthogonal projector onto the range of B∗.
If σ is an exact solution of system (5), then u in (5) can be
found from the equationB∗u = −Aσ+g ∈ R(B∗). If σ is an
approximate solution of system (5) such that the condition
Aσ − g ∈ R(B∗), which is necessary and sufficient for the
existence of u, does not hold, then u can be computed from
the projected equation B∗u = P (−Aσ + g) ∈ P. Both the
original and the projected equations for u are wellposed by
the LBB assumption, i.e. R(B∗) = P and
‖u‖V/N(B∗) ≤
‖σ‖H +
‖g‖H.
Whether the LBB assumption is necessary for wellposed-
ness of the equation for u depends on if the set of all possible
right-hand sides g − Aσ gives the whole subspace R(B∗),
see [10]. For example, in a practically important case g = 0
we have B∗u = −Aσ = −PAσ ∈ R(PA) ⊆ R(P ). If the
latter inclusion is strict, it opens up an opportunity for a
weaker, compared to the original LBB, assumption of well-
posedness of the above equation for u.
In the present paper, however, we are concerned with
finding σ, not u. The LBB condition for the bilinear form
b appears to be of no importance for our results in the
next section where we analyze wellposedness of system (5)
with respect to the σ unknown only, assuming that the u
unknown is of no interest, or can be found for a given σ
using some postprocessing.
4. Coercivity conditions
4.1. The standard coercivity condition
We finally get to the main topic of the paper: an assump-
tion on A which is a condition of wellposedness of (5) with
respect to σ. For the reader’s convenience, we briefly repeat
the necessary notation and the system of equations for σ
to make this section self-consistent. LetH be a real Hilbert
space and P be an orthoprojector in H with a null-space
N(P ) = N and a rangeR(P ) ≡ P—we emphasize that the
range of any orthoprojector in a Hilbert space is closed. Let
A be a linear and bounded operator such that 0 ≤ A∗ = A
on H. The last two lines in system (6) represent an oper-
ator form of system (4); they do not involve the Lagrange
multiplier u or the operator B and determine the primary
unknown σ ∈ H:
P⊥(Aσ − g) = 0 in H,
P (σ − f) = 0 in H,
where g ∈ H and f ∈ H are given and P⊥ ≡ I−P.We can
also replace system (8) with the following equivalent single
equation:
P⊥A |N ψ = P⊥g − P⊥APf ∈ N, σ = ψ + Pf, (9)
where in (9) we take a restriction of the operator P⊥A on
its invariant closed subspace N, and we are looking for a
solutionψ ∈ N. Then the necessary and sufficient condition
of wellposedness of problem (9) for an arbitrary g ∈ H is,
clearly, that the range of P⊥A |N is N. This leads to the
traditional assumption, see [10], a(σ, σ) ≥ ca > 0, ∀σ ∈
N, ‖σ‖H = 1 or, in an operator form, A ≥ caI on N ⊆ H,
since A is selfadjoint nonnegative. Thus, this assumption
is also necessary and sufficient [9, 10] for wellposedness of
system (5) with respect to σ for an arbitrary g ∈ H. In
the rest of the section, we analyze the scenario, where A
is selfadjoint nonnegative on H, but may be degenerate
on N, so we impose necessary restrictions on g ∈ H, and
determine a generalized coercivity condition that covers the
case of the degeneracy.
4.2. Existence, uniqueness, and wellposedness
Before we investigate the existence and uniqueness of the
solution σ, we prove the following technical, but important,
lemma.
Lemma 4.1 Let P be an orthoprojector in H with a null-
space N(P ) = N and a range R(P ) ≡ P = N⊥, and A be
a linear and bounded operator such that 0 ≤ A∗ = A on H.
N(P⊥A) ∩N = N(A) ∩N, (10)
{N(P⊥A) ∩N}⊥ = R(A) +P, (11)
N(P⊥A) ∩N
⊕ P⊥R(A). (12)
Proof.We first verify (10). It follows fromN(P⊥A) ⊇ N(A)
that the right-hand side of (10) is included in the left-hand
side. To prove the reverse inclusion, let ϕ ∈ N and P⊥Aϕ =
0, then 0 = (P⊥Aϕ,ϕ) = (Aϕ,ϕ) = ‖A1/2ϕ‖2 (recall that
A ≥ 0). Then A1/2ϕ = 0 and Aϕ = 0. Therefore, equality
(10) holds.
Equality (11) follows from (10), by substituting N(A) =
F⊥ and R(A) = F in the well-known simple identity
F⊥ ∩ P⊥ = (F + P)⊥ and noting that (R(A) + P)⊥⊥ =
R(A) +P = R(A) +P by properties of the closure.
Finally, to obtain the second term in the orthogonal de-
composition (12) of N we see that by (11) {N(P⊥A) ∩
N}⊥ ∩N = R(A) +P ∩N; at the same time
R(A) +P ∩N= P⊥R(A) +P ∩N
P⊥R(A)⊕P
∩N = P⊥R(A),
which completes the proof of the lemma. ✷
We start with the solution uniqueness.
Lemma 4.2 Suppose that for some fixed g ∈ H and f ∈ H
there exists a solution σ of (8). Then it is unique provided
that N(A) ∩N = {0};otherwise, all possible solutions yield
the hyperplane σ+ {N(A)∩N} and there exists the unique
normal (with minimal norm in H) solution of (8) that can
be also defined as a common element of the above hyperplane
and the closed subspace R(A) +P, which is the set of all
normal solutions for all possible f and g.
Proof. All solutions of (8) with g = f = 0 constitute
the closed subspace N(P⊥A) ∩N(may be 0-dimensional),
which by (10) is the same as N(A) ∩ N. Hence, all solu-
tions of (8) with the given g and f, provided that there
exists at least one solution σ, constitute the hyperplane
σ + N(A) ∩ N. It is known that each closed hyperplane
in a Hilbert space has a unique element with the minimal
norm, i.e. the element that is orthogonal to the directing
closed subspace N(A)∩N of the hyperplane. The orthogo-
nal complement to the directing closed subspace is already
given by (11). ✷
In the rest of the subsection we use the following equation
equivalent to (9):
(P⊥A+ P )σ = P⊥g + Pf. (13)
The assumptions on the right-hand side of the system (8)
which ensure the existence of a solution are rather standard
and follow from (13) easily.
Lemma 4.3 For any f ∈ H there exists a solution of (8) if
and only if g ∈ R(A)+P, i.e. P⊥g+Pf ∈ P⊥R(A)+P =
R(A) +P.
Proof. The subspace (not necessarily closed) P⊥R(A) +P
is simply the range of the operator P⊥A + P of equation
(13). ✷
The subspaceR(A)+P that appears in Lemmas 4.2 and
4.3 plays the central role in the following necessary and
sufficient conditions of wellposedness.
Theorem 4.1 The following statements are equivalent:
(i) The subspace R(A) +P is closed.
(ii) The subspace AN+P is closed.
(iii) The subspace P⊥R(A) is closed.
(iv) The subspace P⊥AN is closed.
(v) Problem (13) with f ∈ H and g ∈ R(A) +P is well-
posed in the factor-space, ‖σ‖H/{N(A)∩N} ≤ c(‖g‖ +
‖f‖), or, equivalently, ‖σ‖ ≤ c(‖g‖ + ‖f‖) for the
normal solution σ ∈ R(A) +P.
Proof. (1)⇔(3) We have R (A)+ P= P⊥R(A)⊕P.
(1)⇔(2) The subspace P⊥R(A) ⊕ P = R(A) + P is the
range of the operator P⊥A + P. The range of a bounded
operator is closed if and only if the range of the conjugate
operator is closed.
(2)⇔(4) Using the same arguments as above, AN + P =
P⊥AN⊕P.
(1)⇔(5) The operator P⊥A + P is bounded and defined
everywhere on a Hilbert space, thus it is closed. Therefore,
the (pseudo)inverse operator
(P⊥A+ P )−1 : R(P⊥A+ P ) → H/{N(A) ∩N}
is closed. It is bounded if and only if its domain of definition
R(P⊥A + P ) is closed. A normal solution is a convenient
representant of a factor-class in a Hilbert space. ✷
4.3. Generalized coercivity conditions
Statements (1)–(4) in Theorem 4.1 may not be so easily
verifiable in practice, so we want to find a somewhat eas-
ier assumption that generalizes the standard coercivity as-
sumption A ≥ caI on N ⊆ H, which itself does not hold if
the operatorA vanishes on a nontrivial subspace ofN ⊆ H.
Let us return back to equation (9). We remind the reader
that the first equation in (8) is equivalent to the orthog-
onal expansion σ = ψ + Pf, where ψ = P⊥σ ∈ N. This
and the second equation in (8) lead to (9) that we present
here, introducing a special notation K = P⊥A |N, in the
equivalent form
Kψ = φ, ψ ∈ P⊥R(A), φ = P⊥g − P⊥APf ∈ P⊥R(A).(14)
under the assumption that g ∈ R(A) +P.
The operatorK is bounded, selfadjoint, and nonnegative
definite on N, where N ⊆ H inherits the scalar product
and the norm of H, so there exists a bounded, selfadjoint,
and nonnegative definite square root
K on N. Applying
the inf-sup condition to the operator
K on N, by direct
analogy with Lemmas 3.1 and 3.2 and their proofs, we have
that N
= N(K) and
Theorem 4.2 The following statements are equivalent:
(i) The subspace R
⊆ N is closed.
(ii) The subspace R(K) ⊆ N is closed.
(iii) The inf-sup condition for the operator
K on N
Kǫ, σ
‖ǫ‖N/N(K)‖σ‖N
= inf
‖ǫ‖N/N(K)
> 0 (15)
holds.
(iv) The norm of the operator K−1 : R(K) → N/N(K)
is equal to ρ <∞.
Moreover, under either of the assumptions we have
= R(K).
Noticing that R(K) = P⊥AN, we immediately see that
statements (4) in Theorem 4.1 and (2) in Theorem 4.2 are
the same, so all statements of Theorems 4.1 and 4.2 are
equivalent. Our last goals in this subsection are to present
statement (3) of Theorem 4.2 in original terms, so that it
resembles the coercivity condition, and to bound the norm
of the solution in terms of the norms of the right-hand sides,
using statement (4) of Theorem 4.2.
Theorem 4.3 For any g ∈ R(A) + P the following as-
sumption
A ≥ 1
I on the subspace P⊥R(A) (16)
with a (finite) constant ρ > 0 is necessary and sufficient for
the normal solution σ with P⊥σ ∈ P⊥R(A) to exist and
to be unique and continuous in f ∈ H and g ∈ R(A) + P.
Moreover, assumption (16) implies
‖σ‖2 ≤ ‖f‖2 + ρ2‖g −APf‖2. (17)
Proof. First, we note that inequality (16) on the subspace
P⊥R(A) is equivalent to the same inequality on its clo-
sure P⊥R(A) because of the continuity of A and the scalar
product. Second, as (ǫ,Kǫ) = (ǫ, P⊥Aǫ) = (ǫ, Aǫ) for all
ǫ ∈ P⊥R(A) ⊆ N, inequality (16) is also equivalent to
I on the closed subspace P⊥R(A). (18)
Now we show that (18) is equivalent to (15), which is
condition (3) of Theorem 4.2. For the numerator in (15),
we have
= (ǫ,Kǫ). To handle the denominator in
(15), we remind the reader the orthogonal decomposition
N(P⊥A) ∩N
⊕P⊥R(A) stated as (12) and proved
in Lemma 4.1. Splitting ǫ ∈ N according to this orthogo-
nal decomposition, we see that its first component—from
N(P⊥A) ∩ N = N(K)—vanishes both in the numerator,
since it is in the null-space of K, and in the denominator
of (15), by the definition of the factor-norm, which gives
(18), where only the second component—from P⊥R(A)—
survives.
We conclude that (16) is equivalent to (15), which is
condition (3) of Theorem 4.2, and thus, to all statements
of Theorems 4.1 and 4.2. Finally, if (16) holds then the
subspace R(P⊥A) is closed, the operator K : P⊥R(A) 7→
P⊥R(A) is an isomorphism and problem (14) is wellposed
for f ∈ H and g ∈ R(A) +P, i.e.
‖ψ‖ ≤ ρ‖P⊥g − P⊥APf‖ ≤ ρ‖g −APf‖ (19)
by Theorem 4.2. Estimate (17) follows from σ = ψ + Pf
and (19) due to the statement of Lemma 4.2 that the nor-
mal solution σ ∈ R(A) +P, that is, ψ ∈ P⊥R(A) =
P⊥R(A) +P = P⊥ ∩ (P⊥ ∩N(A))⊥ is the corresponding
part of the orthogonal expansion σ = ψ + Pf for the nor-
mal solution. ✷
4.4. Minimum gap between subspaces
The rest of the section concerns the case where the range
of A is closed, so assumption (16) can be equivalently re-
formulated using the minimum gap between some relevant
subspaces. We first find a simple way to check if the range
of A is closed.
Lemma 4.4 Condition
A ≥ 1
I on the subspaceR(A) ≡ D (20)
with a (finite) constant ρD > 0 is equivalent to closedness
of D.
Proof. The operator A is a linear, bounded, and every-
where defined. Thus, it is closed and its inverse A−1 : D →
H/N(A) is also closed. Boundedness of the inverse is equiv-
alent, on the one hand, to condition (20) and, on the other
hand, to closedness of D. ✷
Now we are ready to present a simplified version of the
necessary and sufficient condition of wellposedness (16),
assuming that the range of A is closed.
Theorem 4.4 Let the range R(A) ≡ D be closed, the or-
thoprojector onD be denoted byD, and the constant ρD > 0
be defined by (20). Then inequality (16) is equivalent to the
inequality
κ ≡ inf
ψ∈P⊥D
> 0. (21)
In particular, (20) and (21) lead to (16) with ρ = ρD/κ
Proof. We have R(P⊥A) = P⊥R(A) = P⊥D, i.e. the
subspaces indicated in (16) and (21) coincide. Now the
main assertion of the Lemma is a consequence of relations
(Dψ,Dψ) ≤ (Aψ,ψ) = (ADψ,Dψ) ≤ ‖A‖(Dψ,Dψ)
which hold for an arbitrary ψ ∈ H. ✷
The next two lemmas provide alternative assumptions,
equivalent to (21), which are necessary and sufficient for
wellposedness, assuming that the range of A is closed. It is
important to have a choice of a criterion that may be easier
to check in a practical application. For aesthetic reasons we
denote N ≡ P⊥.
Lemma 4.5 Let D and P be orthogonal projectors onto
closed subspaces D and P, and let D⊥ = I −D and P⊥ =
I − P be orthogonal projectors onto the orthogonal comple-
ments D⊥ and P⊥, respectively. The following statements
are equivalent:
(i) The subspace P⊥D is closed.
(ii) The subspace D+P is closed.
(iii) The subspace D⊥ +P⊥ is closed.
(iv) The subspace PD⊥ is closed.
Proof. (1)⇔(2) The subspace P⊥D is closed iff the sub-
space P⊥D⊕P = D+P is closed as the terms are orthog-
onal in the first expression.
(2)⇔(3) By Theorem IV-4.8 of [12], a sum of closed sub-
spaces in a Hilbert space is closed if and only if the sum of
their orthogonal complements is closed.
(3)⇔(4) Using the same arguments as above, P⊥ +D⊥ =
P⊥ ⊕ PD⊥. ✷
Lemma 4.6 Using the notation of Lemma 4.5, the follow-
ing equalities hold:
ψ∈P, ψ 6∈D
dist{ψ;D}
dist{ψ;D ∩P}
= inf
ψ∈D, ψ 6∈P
dist{ψ;P}
dist{ψ;P ∩D}
= inf
ψ∈D⊥, ψ 6∈P⊥
dist{ψ;P⊥}
dist{ψ;P⊥ ∩D⊥}
= inf
ψ∈P⊥, ψ 6∈D⊥
dist{ψ;D⊥}
dist{ψ;D⊥ ∩P⊥}
= inf
ψ∈P⊥D
= inf
ψ∈D⊥P
= inf
ψ∈PD⊥
‖D⊥ψ‖
= inf
ψ∈DP⊥
‖P⊥ψ‖
Moreover, each statement in the previous Lemma is equiv-
alent to the positiveness κ > 0 in (21).
Proof. The first three equalities are derived in Section IV-4
of [12] on the minimum gap between subspaces, along with
a statement that positiveness of the minimum gap between
two given subspaces is a necessary and sufficient condition
of the sum of the subspaces, in our case,D+P, to be closed.
We now prove that
ψ∈P⊥, ψ 6∈D⊥
dist{ψ;D⊥}
dist{ψ;D⊥ ∩P⊥}
= inf
ψ∈P⊥D\{0}
All other equalities can be then trivially derived from the
previous ones just by interchanging P and D.
We first notice that in the right-hand side we can apply
the inf to the closure P⊥D \ {0} as well, because a norm is
a continuous function,
ψ∈P⊥D\{0}
‖ψ‖ = infψ∈P⊥D\{0}
‖ψ‖ .
We have, P⊥D = P⊥ ∩ (P⊥ ∩D⊥)⊥ as N(DP⊥) = P ⊕
(P⊥ ∩D⊥). The latter can be checked directly.
We always have dist{ψ;D⊥} = ‖Dψ‖. If ψ ∈ P⊥D =
P⊥∩(P⊥∩D⊥)⊥ ⊆ (P⊥∩D⊥)⊥,we also have dist{ψ;D⊥∩
P⊥} = ‖ψ‖. Thus,
dist{ψ;D⊥}
dist{ψ;D⊥ ∩P⊥}
, ψ ∈ P⊥D \ {0}.
Finally, using the orthogonal representation P⊥ = (P⊥ ∩
D⊥)⊕P⊥D, everyϕ ∈ P⊥ can be written as the orthogonal
sum ϕ = (ϕ−ψ)⊕ψ, where ϕ−ψ ∈ P⊥ ∩D⊥, ψ ∈ P⊥D.
Then dist{ψ;D⊥} = dist{ϕ;D⊥} and also dist{ψ;D⊥ ∩
P⊥} = dist{ϕ;D⊥ ∩P⊥}; so the value of the ratio
dist{ψ;D⊥}
dist{ψ;D⊥ ∩P⊥}
dist{ϕ;D⊥}
dist{ϕ;D⊥ ∩P⊥}
does not depend on ϕ−ψ and its two infimum values, taken
with respect to ψ ∈ P⊥D \ {0} and ϕ ∈ P⊥, ϕ 6∈ D⊥,
coincide. ✷
Finally, we notice that g = 0 if we apply a saddle point
approach to diffusion or linear elasticity equations. Indeed,
in the Hellinger–Reissner formulation of nonhomogeneous
Lamé equations, our σ represents the stress tensor, the La-
grange multiplier u is the displacement, and if we also in-
troduce the stain ǫ by the stain-displacement relation ǫ =
−B∗u, then the first line in system (5) becomes Aσ − ǫ =
g, which is the constitutive equation (3-D Hooke’s law),
where of course g = 0. The second line in (5) is the equi-
librium equation, where all body and traction forces are
represented by f 6= 0. The assumption g = 0 allows us to
look for even weaker conditions of wellposedness that we
plan to investigate in the future.
Acknowledgments The author thanks Ivo Babuška and
Franco Brezzi for discussions. This work has been stimu-
lated by collaboration with Nikolai S. Bakhvalov. The au-
thor thanks an anonymous referee, who has made numer-
ous useful suggestions to improve the original version of
the paper, and CU-Denver students Donald McCuan and
Christopher Harder for proofreading the paper.
References
[1] D. N. Arnold and R. Winther. Mixed finite elements
for elasticity. Numer. Math., 92(3):401–419, 2002.
[2] I. Babuška and A. K. Aziz. Survey lectures on
the mathematical foundations of the finite element
method. In The mathematical foundations of the fi-
nite element method with applications to partial differ-
ential equations, pages 1–359. Academic Press, New
York, 1972. With the collaboration of G. Fix and R.
B. Kellogg.
[3] N. S. Bakhvalov and A. V. Knyazev. A new iterative
algorithm for solving problems of the fictitious flow
method for elliptic equations. Soviet Math. Doklady,
41(3):481–485, 1990.
[4] N. S. Bakhvalov and A. V. Knyazev. Fictitious domain
methods and computation of homogenized properties
of composites with a periodic structure of essentially
different components. In Gury I. Marchuk, editor,
Numerical Methods and Applications, pages 221–276.
CRC Press, Boca Raton, 1994.
[5] N. S. Bakhvalov and A. V. Knyazev. Preconditioned
iterative methods in a subspace for linear algebraic
equations with large jumps in the coefficients. In
D. Keyes and J. Xu, editors, Domain Decomposition
Methods in Science and Engineering, volume 180 of
Contemporary Mathematics, pages 157–162.American
Mathematical Society, Providence, 1994.
[6] N. S. Bakhvalov, A. V. Knyazev, and G. M. Kobel’kov.
Iterative methods for solving equations with highly
varying coefficients. In Roland Glowinski, Yuri A.
Kuznetsov, Gérard A. Meurant, Jacques Périaux, and
Olof Widlund, editors, Fourth International Sympo-
sium on Domain Decomposition Methods for Partial
Differential Equations, pages 197–205, Philadelphia,
PA, 1991. SIAM.
[7] N. S. Bakhvalov, A. V. Knyazev, and R. R.
Parashkevov. Extension theorems for Stokes and Lame
equations for nearly incompressible media and their
applications to numerical solution of problems with
highly discontinuous coefficients. Numerical Linear
Algebra with Applications, 9(2):115–139, 2002.
[8] M. Benzi, G. H. Golub, and J. Liesen. Numerical solu-
tion of saddle point problems. Acta Numer., 14:1–137,
2005.
[9] F. Brezzi. On the existence, uniqueness and approx-
imation of saddle point problems arising from La-
grangian multipliers. RAIRO Anal. Numer., 2:129–
151, 1974.
[10] F. Brezzi and M. Fortin. Mixed and Hybrid Finite
Element Methods. Springer–Verlag, New York, 1991.
[11] P. Ciarlet, Jr., J. Huang, and J. Zou. Some observa-
tions on generalized saddle-point problems. SIAM J.
Matrix Anal. Appl., 25(1):224–236, 2003.
[12] T. Kato. Perturbation Theory for Linear Operators.
Springer–Verlag, New–York, 1976.
[13] A. V. Knyazev. Analysis of transmission problems on
Lipschitz boundaries in stronger norms. J. Numer.
Math., 11(3):225–234, 2003.
[14] A. V. Knyazev. Iterative solution of PDEwith strongly
varying coefficients: algebraic version. In R. Beauwens
and P. de Groen, editors, Iterative Methods in Linear
Algebra, pages 85–89, Amsterdam, 1992. Elsevier.
[15] A. V. Knyazev and O. Widlund. Lavrentiev regular-
ization + Ritz approximation= uniform finite element
error estimates for differential equations with rough
coefficients. Mathematics of Computation, 72:17–40,
2003.
[16] O. A. Ladyzhenskaya. The boundary value problems
of mathematical physics, volume 49 of Applied Mathe-
matical Sciences. Springer-Verlag, New York, 1985.
[17] J. Xu and L. Zikatanov. Some observations onBabuška
and Brezzi theories. Numer. Math., 94(1):195–202,
2003.
	Introduction
	Abstract symmetric saddle point problems
	Formulations using bilinear forms
	Operator-based formulations
	Inf-sup or LBB condition
	Coercivity conditions
	The standard coercivity condition
	Existence, uniqueness, and wellposedness
	Generalized coercivity conditions
	Minimum gap between subspaces
