LECTURE # 4:
INF-SUP CONDITION AND
BANACH-NEČAS-BABUŠKA THEOREM
R. LAZAROV
1. Abstract Weak Problems and Their FE Approximations
1.1. Abstract Problem. Now we shall consider the following abstract
problem. Consider the Hilbert spaces V and W with norms generated by the
corresponding inner (scalar) products: (U, V )V for U, V ∈ V and (W, Z)W
for W, Z ∈ W. Next, we introduce the bilinear form A(V, W ) : V × W → R
and the linear form L(W ) : W → R. Now consider the following abstract
problem:
(1)
find U ∈ V such that
A(U, W ) = L(W ) ∀W ∈ W.
The aim of this lecture is to introduce an abstract framework to study this
class of problems and discuss the existence and uniqueness of the solution
and its continuous dependence on the data.
In order to illustrate this concept, we first give a number of examples.
Ex # 1: Take U = u, V = W = H01 (Ω),
Z
Z
A(u, w) :=
∇u · ∇wdx and L(w) =
f wdx,
Ω
Ω
L2 (Ω).
with f (x) given function in
As we know from the previous
lectures this is the weak form of the problem −∆u = f for x ∈ Ω
and u(x) = 0 for x ∈ ∂Ω.
Ex # 2: (indefinite elliptic problem). Take U = u, V = W = H01 (Ω),
Z
Z
2
A(u, w) := (∇u · ∇w − ω uw)dx and L(w) =
f wdx
Ω
Ω
L2 (Ω).
with f (x) given function in
This is the weak form of the
Helmholtz equation −∆u−ω 2 u = f for x ∈ Ω with Dirichlet boundary conditions u(x) = 0 for x ∈ ∂Ω.
Ex # 3: (indefinite mixed problem corresponding to second order differfential
equations). Take U = (q, u), W = (r, w), V = W = H(div; Ω) ×
L2 (Ω) and
Z
Z
Z
A(U, W ) :=
q · rdx −
u∇ · rdx −
∇qwdx,
Ω
Ω
Date: April 16, 2013.
1
Ω
2
R. LAZAROV
and
Z
L(W ) := l(w) =
f wdx.
Ω
This is the weak form of the mixed system corresponding to example
#4 considered in Lecture # 0 with K(x) = I, c(x) = 0.
Ex # 4: Stokes system (example # 13 of Lecture # 0) in a weak form. In
this case we set U = (u, p), V = W = H01 (Ω) × L20 (Ω) (with L20 (Ω)
functions in L2 (Ω) with zero mean value on Ω) and seek a pair
(u, p) ∈ H01 (Ω) × L20 (Ω) that satisfies the integral identity
A(U, W ) := A(u, p; v, q) := (∇u, ∇v) − (p, ∇ · v) − (∇ · u, q) = (f, v)
for all (v, q) ∈ H01 (Ω) × L20 (Ω).
Now we introduce the main tool for our analysis. This is not the most
general form, but it is enough for our purposes. The most general theorem
is valid for general Banach spaces.
Theorem 1. (inf-sup) (Banach-Nečas-Babuška, Ladyzhenskaya-BabuškaBrezzi, Babuška-Brezzi) Let V and W be Hilbert spaces. The problem (1) is
well-posed if and only if:
(BNB1)
∀U ∈ V
A(U, W ) = 0 ∀W ∈ W implies U = 0;
(BNB2)
∃α0 > 0 :
sup
W ∈W
A(U, W )
≥ α0 kU kV
kW kW
∀U ∈ V.
Moreover, the following a priori estimate holds
kU kV ≤
1
L(w)
1
sup
:=
kLkW 0 .
α0 W ∈W kW kW
α0
Here kLkW 0 is the norm of L in the dual W 0 of W.
We are not going to prove this theorem but we shall illustrate how to use
it for various boundary value problems for PDEs.
First, we note that if the bilinear form is defined on V × V (in this case
V = W) and is coercive in V then it satisfies the conditions of BNB Theorem.
Indeed, coercivity in V means that
A(U, U ) ≥ α0 kU k2V ,
α0 = const > 0.
Then (BNB1) follows immediately: if A(U, W ) = 0 ∀W ∈ W holds then
choosing W = U and using the coercivity we get 0 = A(U, U ) ≥ α0 kU kV ≥ 0
implies that kU kV = 0 and therefore U = 0.
The inf-sup condition (BNB2) follows easily as well. By choosing W = U
we get
α0 kU k2V
A(U, U )
A(U, W )
≥
≥
= α0 kU kV .
sup
kU kV
kU kV
W ∈V kW kV
LECTURE # 4: INF-SUP CONDITION AND STABILITY
3
1.2. Abstract Galerkin Method. Now we shall consider the Galerkin
method for the problem (1). Let Vh ⊂ V and Wh ⊂ W be finite dimensional.
Later we shall give particular examples of such spaces for the case of second
order elliptic problem. Then the abstract Ritz-Galerkin method is:
(Vh )
find Uh ∈ Vh such that
A(Uh , W ) = L(W )
Now we introduce the discrete inf-sup condition:
A(Uh , W )
(BNB2h )
∃α0∗ > 0 :
sup
≥ α0∗ kUh kV
W ∈Wh kW kW
∀W ∈ Wh .
∀Uh ∈ Vh .
Remark 1. Note that since w ∈ Wh the supremum is taken in a much
smaller set and the discrete inf-sup condition (BNB2h ) does not follow from
the infinite dimensional case (BNB2). For each particular choice of the
spaces Vh and Wh we have to verify the condition (BNB2h ).
Now we can prove the Cea’s Lemma for discrete problems with indefinite
bilinear forms.
Lemma 1. (Cea’s lemma) Assume the discrete inf-sup condition (BNB2h )
is satisfied. Then the abstract Galerkin method (Vh ) has unique solution
Uh ∈ Vh .
If in addition the bilinear form A(U, W ) is continuous with respect to both
U ∈ V and W ∈ W, i.e. there is a constant c0 such that
(2)
A(U, W ) ≤ c0 kU kV kW kW ,
then this solution satisfies the estimate
c0 (3)
kU − Uh kV ≤ 1 + ∗ inf kU − V kV .
α0 V ∈Vh
Proof. We first note that the finite dimensional problem (Vh ) is a square system of linear algebraic equations for the unknown degrees of freedom. Then
a discrete variant of the condition (BNB1) is equivalent to non-singularity
of the corresponding matrix. The discrete variant of the inf-sup condition
(BNB2) is equivalent to the non-singularity of the transposed matrix. But
as we know, for square matrices non-singularity of the matrix is equivalent
to the non-singularity of its transposed. Therefore in the finite dimensional
case we need only the inf-sup condition. Note that inf-sup conditions ensures more than the non-singularity of the matrix, it guarantees that the
lowest eigenvalue is bounded away from 0 uniformly in h.
Since the matrix of the corresponding system is non-singular, the problem
(Vh ) has unique solution Uh . Also Wh ⊂ W and in (1) we can take W ∈ Wh
to get
A(U, W ) = L(W )
and A(Uh , W ) = L(W ) ∀W ∈ Wh ,
which leads to the well-known Galerkin orthogonality
A(U − Uh , W ) = 0
∀W ∈ Wh .
4
R. LAZAROV
Now let V be an arbitrary function in Vh . Then by the discrete inf-sup
condition (BNB2h ) and by Galerkin orthogonality we have
α0∗ kUh − V k ≤ sup
W ∈Wh
A(Uh − V, W )
A(U − V, W )
= sup
.
kW kW
kW kW
W ∈Wh
The continuity (2) ensures that A(U − V, W ) ≤ c0 kU − V kV kW kW and
therefore
α0∗ kUh − V k ≤ c0 kU − V kV for any V ∈ Vh .
Now by triangle inequality and the above estimate we get for any V ∈ V:
c0 kU − Uh kV ≤ kU − V kV + kV − Uh kV ≤ 1 + ∗ kU − V kV .
α0
Taking infimum in V ∈ Vh we get the desired result.
2. Application to Indefinite Elliptic Problems
Now let us illustrate the use of this theorem on Example 2 by choosing
Ω = (0, 1) and taking ω = 5. Then the problem is: find u ∈ H01 (0, 1) such
that
Z 1
Z 1
0 0
A(u, w) :=
(u w − 25uw)dx =
f w dx ∀w ∈ H01 (0, 1).
0
0
Obviously, taking u(x) = sin(πx) (which is obviously in H01 (0, 1) we get
R1
A(u, u) = 0 (π 2 cos2 (πx) − 25 sin2 (πx))dx < 0. Therefore, the form is not
coercive in H01 (0, 1). However, this problem is well posed, a result that will
follow from BNB Theorem.
The condition (BNB1) is obviously satisfied since A(u, w) = 0 for all
w ∈ H01 (0, 1) is equivalent to −u00 (x) = 25u and u(0) = u(1) = 0. We
know that the problem −u00 (x) = λ2 u and u(0) = u(1) = 0 has infinitely
many solution un = sin(λn x), λn = πn, n = 1, 2, . . . called eigenpairs, and
these are all nontrivial solutions of the problem. But 5 6= λn for any n and
therefore the above problem must have the solution u(x) ≡ 0, this means
that the condition (BNB1) is satisfied.
More difficult is to show the inf-sup condition (BNB2). This we shall do
in the more general case of indefinite problems. This will follow from the
more general result we shall state below.
Theorem 2. Assume that the bilinear form A(u, v) defined on H01 (Ω) ×
H01 (Ω) satisfies:
(a) Görding inequality, i.e. for some constants α > 0 and β > 0
(4)
A(u, u) ≥ αkuk2H 1 − βkuk2L2
∀u ∈ H01 ;
(b) if u ∈ H01 (Ω) satisfy A(u, v) = 0 ∀v ∈ H01 (Ω) then u ≡ 0.
Then there is a constant α0 > 0 such that
A(u, v)
(5)
sup
≥ α0 kukH01 .
1
v∈H kvkH01
0
LECTURE # 4: INF-SUP CONDITION AND STABILITY
5
Before proving this theorem let us apply it to the Helmholtz problem of
Example # 2 with A(u, v) = (∇u, ∇v) − ω 2 (u, v). Then Görding inequality
follows immediately
Z
Z
2
2
2
A(u, u) ≥ (|∇u| + u )dx − (ω + 1) u2 dx = kuk2H 1 − (ω 2 + 1)kuk2L2 .
Ω
Ω
If ω 2 is not an eigenvalue of the operator −∆ with homogeneous Dirichlet
boundary conditions, then A(u, v) = 0 for all v ∈ H01 implies u ≡ 0. Thus,
Helmholtz problem for ω 2 not an eigenvalue of −∆ has weak formulation
with a bilinear form that satisfies the inf-sup condition.
Proof. Assume that (5) does not hold. This means that for any n we can
find un such that
sup
v∈H01
A(un , v)
1
≤ kun kH01 .
kvkH01
n
Obviously we can take kun kH01 = 1. Thus, we have a sequence {un }∞
n=1 ,
1
1
kun kH 1 = 1. Since {un } is bounded in H and the space H is compactly
embedded in L2 we can extract a sub-sequence {uk }∞
k=1 that converges in
L2 (Ω) to its limit u0 ∈ L2 (Ω). Then
0 ≤ sup
v∈H01
A(un , v)
=
sup
kvkH01
v∈H 1 ,kvk
0
A(un , v) ≤
H 1 =1
1
.
n
This implies 0 ≤ A(un , v) ≤ n1 for any v ∈ H 1 and A(un , un ) → 0 when
n → ∞. Taking the limit in the first inequality we get
0 = lim A(un , v) = A(u0 , v)
n→∞
∀v ∈ H01 .
By assumption (b) u0 = 0 as an element in L2 (Ω) and therefore ku0 kL2 = 0.
However, by Görding inequality (4) we have
1
≥ A(un , un ) ≥ αkun k2H 1 − βkun k2L2 .
n
By taking limit when n → ∞ and using the following facts kun kH 1 = 1, and
limn→∞ kun kL2 = 0 we get an obvious contradiction
0 = lim A(un , un ) ≥ α > 0.
n→∞
This completes the proof.
Remark 2. We remark that every second order elliptic problem falls into
the class of probems covered by the above Theorem as long as the zero is not
an eigenvalue of the corresponding operator.
6
R. LAZAROV
3. FEM for Indefinite Elliptic Problems
3.1. Problem Formulation. Now we consider the general second order
elliptic problem
(V )
find u ∈ H01 (Ω) such that A(u, v) = L(v), ∀ v ∈ H01 (Ω) ,
where
A(u, v) =
Z Z
f vdx.
K(x)∇u · ∇v − ub(x) · ∇v + c(x)uv dx, L(v) =
Ω
Ω
Recall that we have assumed that K(x) is a symmetric uniformly in Ω
positive definite matrix. Then it is easy to show that the bilinear form
satisfies Görding inequality (this is left as an exercise). We assume that
A(u, v) = 0 for all v ∈ H01 (Ω) implies u = 0. Then according to Theorem 2
the problem (V ) has unique solution.
Now we consider FEM for this problem using linear finite elements over
a partition of the domain Ω into triangles. Let Vh ⊂ H01 (Ω) be the corresponding finite element space. Then the FEM for the problem (V) is:
(Vh )
find uh ∈ Vh such that A(uh , v) = L(v), ∀ v ∈ Vh .
The main question is whether this problem is well posed and if so, what is
the error bound for the FE solution.
To simplify the exposition we note that Vh ⊂ H01 (Ω) := V and it inherits
the norm in H01 so that kuh kV = kuh kH 1 . Now we prove the following
theorem:
Theorem 3. If h ≤ h0 , where h0 is sufficiently small (depending on Ω,
coefficients of the differential equation, etc), then there is an independent of
h constant α0∗ , such that the discrete inf-sup condition is valid, namely
(6)
∀uh ∈ Vh :
sup
v∈Vh
A(uh , v)
≥ α0∗ kuh kV .
kvkV
Proof. For the proof we shall need the following facts:
(1) the bilinear form A(·, ·) is continuous in V, i.e. it satisfies the estimate
A(v, w) ≤ CkvkH 1 kwkH 1 , ∀v, w ∈ V;
(2) Ritz-projection Rh v ∈ Vh of a function v ∈ V defined as the solution
to the FE problem (∇Rh v, ∇φ) = (∇v, ∇φ), ∀φ ∈ Vh is stable in
H 1 , that is
c1 kRh vkH 1 ≤ kvkH 1 ,
with c1 > 0,
and approximates v in L2 -norm by satisfying
kv − Rh vkL2 ≤ ChkvkH 1 ,
with C > 0.
LECTURE # 4: INF-SUP CONDITION AND STABILITY
7
Now since uh ∈ H 1 we use the inf-sub condition established in Theorem 2
an proceed as follows
A(uh , v)
A(uh , v − Rh v)
A(uh , Rh v)
α0 kuh kH 1 ≤ sup
≤ sup
+ sup
.
kvkH 1
kvkH 1
v∈H 1 kvkH 1
v∈H 1
v∈H 1
0
0
0
By using the definition of the Ritz projection, (∇(Rh u − u), ∇v) = 0 for all
v ∈ Vh , the continuity of the bilinear form, and the approximation property
of the Ritz projection, that is kRh v − vkL2 ≤ chkvkH 1 , for the first term we
get
sup
v∈H01
A(uh , v − Rh v)
−ω 2 (uh , v − Rh v)
= sup
kvkH 1
kvkH 1
v∈H 1
0
≤ sup
v∈H01
Ckuh kH 1 kv − Rh vkL2
≤ Chkuh kH 1 .
kvkH 1
For the second term we use the stability of Ritz-projection to get
A(uh , Rh v)
A(uh , Rh v)
1
A(uh , v)
sup
≤ sup
=
sup
kvkH 1
c1 v∈Vh kvkH 1
v∈H 1
v∈H 1 c1 kRh vkH 1
0
0
As a result we get
(α0 − Ch)kuh kH 1 ≤
1
A(uh , v)
sup
,
c1 v∈Vh kvkH 1
which produces the desired result for sufficiently small h. For example, if
α0
h0 = 2C
we get α0∗ = c12α0 .
Remark 3. We have used the estimate kv − Rh vkL2 ≤ ChkvkH 1 , which
is valid if the solution of the elliptic problem (∇v, ∇φ) = (f, v) has full
regularity. This means that u ∈ H 2 (Ω). This is the case of convex domain
Ω with polygonal boundary. For non-convex domains however, the estimate
is not valid. But in this case we can prove an estimate kv − Rh vkL2 ≤
1
Ch 2 kvkH 1 , which will also lead to the desired inf-sup condition.
As a result of this theorem we have the following result:
Theorem 4. The solution of the problem (Vh ) exists and satisfies the a
priori estimate
1
kuh kH 1 ≤ ∗ kf kL2 .
α0
3.2. Error Analysis. The error analysis follows in a standard way from
Cea’s Lemma. Now taking v to be the finite element interpolate Πh u ∈ Vh
and using the approximation properties of the interpolate,
(7)
ku − Πh ukV ≤ ChkukH 2
we get the desired error estimate: kuh − ukH 1 ≤ ChkukH 2 .
In this lecture we used the approximation property (7) of the finite element interpolate of a function u ∈ H 2 (Ω). In the next lecture we shall
8
R. LAZAROV
establish this error bound. More precisely, we shall prove the following theorem:
Theorem 5. Assume the triangulation of Ω is shape-regular (we shall define
this rigorously), the maximal diameter of the finite elements is h, and the
finite element space consists of continuous piece-wise linear functions. Then
for u ∈ H 2 (Ω) we have
ku − Πh ukL2 + hk∇(u − Πh u)kL2 ≤ Ch2 kukH 2 (Ω) ,
where Πh u is the nodal finite element interpolate.
4. Exercises
Problem 1: Consider the problem
−∆u + b · ∇u + c(x)u = f,
u = 0,
on Ω,
on ∂Ω.
Here you may take Ω to be the unit square (in R2 ). Assume that c(x) ≥ 0,
the vector field b(x) is smooth and satisfies ∇ · b(x) = 0 and f ∈ L2 (Ω).
(1) Derive a variational formulation of the above problem. You must
identify the space V , the bilinear form A(·, ·), and the linear functional L(·).
(2) Show that the resulting bilinear form is coercive.
Acknowledgments
The author expresses thanks to Prof. S. Dimova for the useful discussions
during the preparation of this lecture and her critical remarks regarding the
content and the exposition level. The support by the Project AComIn ”Advanced Computing for Innovation”, grant # 316087, funded by the European FP7 Capacity Program (Research Potential of Convergence Regions),
is gratefully acknowledged.
References
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Mathematics, v. 40, SIAM, 2002.
[2] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Series of Applied
Mathematical Sciences v. 159, Springer-Verlag, 2004.
[3] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, volume
19, American Mathematical Society, 1991.
[4] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic PDEs,
Springer-Verlag, New Yrok Inc, 2003.
[5] S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods,
Springer-Verlag, Texts in Applied Mathematics 45, 2003.
[6] M. Renardy and R. Rogers, An Introduction to Partial Differential Equations, Texts
in Applied Mathematics 13, Springer-Verlag, 1993.