1
General variational problems and their approximations
1.1
The Lax-Milgram-Nirenberg Lemma
Theorem. Let H be a Hilbert space with inner product (·, ·)H and norm k·kH .
Suppose that the bilinear form B : H ×H → IR satisfies the following conditions.
• Continuity: there is a positive constant C such that
|B(W, V )| ≤ CkW kH kV kH
∀W, V ∈ H.
• The ”inf-sup” condition: there is a positive constant α such that
B(W, V )
≥ αkW kH
V ∈H, V 6=0 kV kH
sup
∀W ∈ H.
• The condition
sup B(W, V ) 6= 0
∀V ∈ H.
W ∈H
Then the variational problem: find U ∈ H such that
B(U, V ) = F (V )
∀V ∈ H,
has a unique solution depending continuously on the data:
kU kH ≤ α−1 kF kH 0 ,
where k·kH 0 is the dual norm
kF kH 0 = sup
V ∈H
F (V )
.
kV kH
Proof: (Cf. I. Babuska [], L.C. Evans. PDE’s pp. 297-9)
Step 1). For every W ∈ H
ΦW (V ) = B(W, V )
defines a continuous linear functional on H. By the Riesz representation theorem
there is Z ∈ H such that
ΦW (V ) = (Z, V )H .
Hence, we have a linear mapping A : H → H , Z = A(W ), such that
(A(W ), V )H = B(W, V ) ∀W, V ∈ H.
From the continuity condition it follows that A is bounded:
kAk ≤ C.
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2). The mapping A is bounded from below and R(A) is closed: We have
kA(W )kH =
sup
(A(W ), V )H =
V ∈H, kV kH =1
sup
B(W, V )H ≥ αkW kH .
V ∈H, kV kH =1
Let A(Wn ) be a Cauchy-sequence. From above we have
kA(Wn ) − A(Wm )kH ≥ αkWn − Wm kH ,
and hence Wn is also Cauchy, converging to (say) W ∈ H. Since A is bounded
we have A(Wn ) → A(W ) which shows that R(A) is closed.
3). R(A) = H. If not, there exists V0 6= 0 such that
(A(W ), V0 )H = 0 ∀W ∈ H.
This is equivalent with
B(W, V0 ) = 0
∀W ∈ H
which contradicts the third condition assumed of B.
4). Next we apply the Riesz representation theorem to the right hand side
as well: there exits G ∈ H such that
F (V ) = (G, V )H .
The variational problem is now equivalent to
(A(U ), V )H = (G, V )H
i.e.
A(U ) = G
with solution
U = A−1 (G).
5). From the inf-sup condition we now finally have
αkU kH ≤
B(U, V )
F (V )
= sup
= kF kH 0 ,
kV
k
kV
kH
H
V ∈H, V 6=0
V ∈H, V 6=0
sup
which also shows the uniqueness.
1.2
Finite Element Discretization
Choose a subspace Hh ⊂ H and solve Uh ∈ Hh from
B(Uh , V ) = F (V ) ∀V ∈ Hh .
We then have the analog to Cea’s lemma.
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Theorem. Suppose that the following discrete ”inf-sup” condition is valid:
there is a constant γ > 0 such that
B(W, V )
≥ γkW kH
V ∈Hh , V 6=0 kV kH
sup
Then it holds
∀W ∈ Hh .
C
) inf kU − ZkH ,
γ Z∈Hh
kU − Uh kH ≤ (1 +
where C is the constant in the continuity condition.
Proof: 1). Testing with v ∈ Hh in the the continuous problem and subtracting from the discrete formulation gives
B(U − Uh , V ) = 0 ∀V ∈ Hh .
2). Let Z ∈ Hh be arbitrary. Choosing W = Uh − Z in the inf-sup then gives
γkUh −ZkH ≤
B(U − Z, V )
B(Uh − Z, V )
=
sup
≤ CkU −ZkH ,
kV
k
kV kH
H
V ∈Hh , V 6=0
V ∈Hh , V 6=0
sup
i.e.
kUh − ZkH ≤
C
kU − ZkH .
γ
The claim now follows from the triangle inequality (and ”taking the inf”).
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2.1
Application to the Stokes problem
The Babuska-Brezzi splitting for Stokes problem
For the Stokes equations we define H := H01 (Ω)N × L20 (Ω), with the norm
k(v, q)k2H := kvk21 + kqk20 ,
and the bilinear form
B((v, q), (z, r)) := (∇v, ∇z) − (div z, q) − (div v, r).
The dual space is now H 0 = H −1 (Ω)n × L20 (Ω).
The inf-sup is now:
sup
(z ,z)∈H
B((v, q), (z, r))
≥ αk(v, q)kH
k(z, r)kH
∀(v, q) ∈ H.
The Babuska-Brezzi theory for mixed (or saddle point) problems says that this
is a consequence of the two conditions.
The Babuska-Brezzi conditions.
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• The ellipticity: There is a constant C1 such that
(∇v, ∇v) ≥ C1 kvk21
∀v ∈ H01 (Ω)n .
• The LBB (Ladyshenskaya-Babuska-Brezzi) condition: there is a positive
constant C2 such that
sup
v
∈H01 (Ω)n
(div v, q)
≥ C2 kqk0
kvk1
∀q ∈ L20 (Ω).
The first estimate above is simply the Poincar´e inequality.
In order to keep track of the influence of the LBB constant it will be convenient to work with the norm k∇vk0 for v ∈ H01 (Ω)n . The LBB we then write
as: there is a positive constant β such that
sup
w
∈H01 (Ω)n
(div w, q)
≥ βkqk0
k∇wk0
∀q ∈ L20 (Ω).
Then we perform the analysis with the ”triple-bar” norm:
|k(v, q)k|2H := k∇vk20 + β 2 kqk20 .
We also use the standard trick, the ”arithmetic-geometric-mean inequality”
(AGM):
1
ε
|ab| ≤ a2 + b2 , a, b ∈ IR, ε > 0.
2
2ε
or more precisely
ε
1
−|ab| ≥ − a2 − b2 ,
2
2ε
a, b ∈ IR,
ε > 0.
Let’s now build the inf-sup for B from the ellipticity and the LBB.
Proof: 1). Let (v, q) ∈ H be arbitrary. We first have
B((v, q), (v, −q)) = (∇v, ∇v) − (div v, q) + (div v, q) = k∇vk20 .
2). The LBB condition reformulated is: there is w ∈ H01 (Ω)n such that
(div w, q) ≥ βkqk20 and k∇wk0 = kqk0 .
Using this, Schwartz and the AGM (with ε = β) gives
B((v, q), (−w, 0)) = −(∇v, ∇w) + (div w, q)
≥ −(∇v, ∇w) + βkqk20
≥ −k∇vk0 k∇wk0 + βkqk20
1
β
≥ − k∇vk20 − k∇wk20 + βkqk20
2β
2
1
β
= − k∇vk20 + kqk20 .
2β
2
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3). For δ > 0 we then have
B((v, q), (v − δw, −q)
= B((v, q), (v, −q)) + δ B((v, q), (−w, 0)
δ δβ
≥
1−
k∇vk20 +
kqk20
2β
2
β2
1
1
k∇vk20 +
kqk20 = |k(v, q)k|2H .
=
2
2
2
when choosing δ = β.
4). For z = v − δw = v − βw and r = −q we thus have
B((v, q), (z, r)) ≥
1
|k(v, q)k|2H
2
and (AGM again)
|k(z, r)k|H ≤ k∇(v − βw)k0 + βkqk0
≤ k∇vk0 + βk∇wk0 + βkqk0
= k∇vk0 + 2βkqk0 ≤ 2 k∇vk0 + βkqk0
√
≤ 2 2|k(v, q)k|H .
5). Combining gives (if I calculated the constants right)
√
B((v, q), (z, r))
2
sup
|k(v, q)k|H ∀(v, q) ∈ H.
≥
|k(z,
r)k|
8
H
(z ,r)∈H
Since k·kH and |k·k|H are equivalent the claim is proved.
2.2
The Babuska-Brezzi condition for the FEM
We discretize Stokes: find (uh , ph ) ∈ V h × Ph =: Hh ⊂ H such that
B((uh , ph ), (v, q)) = (f , v)
∀(v, q) ∈ Hh .
The discrete the inf-sup follows from the ellipticity condition and the discrete
LBB condition. From Cea’s lemma we get the following result.
Theorem. Suppose that the discrete spaces satisfy the condition: there is a
constant β > 0, independent of h, such that
(div w, q)
≥ βkqk0
sup
w∈V h k∇wk0
5
∀q ∈ Ph .
Then there is a constant C > 0, independent of h, such that
ku − uh k1 + kp − ph k0 ≤ C inf ku − vk1 + inf kp − qk0 .
q∈P
h
v ∈V h
The purpose of using the triple-bar norm above was to trace the influence
of the LBB constant. When we redo it for the discrete problem we get the
estimates:
ku − uh k1 ≤ Cβ −1 inf ku − vk1 + inf kp − qk0 .
q∈P
h
v ∈V h
and
kp − ph k0 ≤ Cβ −2
inf ku − vk1 + inf kp − qk0 .
q∈P
h
v ∈V h
These holds also in cases when β > 0 depends on the mesh length h (e.g. the
Q1 –P0 ) and then shows that the accuracy is degenerated for instable methods.
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