Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 132

Suppose condition (iii) is satisfied. Then for given v ∈ V ⊥ , we define a
functional g ∈ V 0

by w −→ (v, w) . Since B is an isomorphism, there exists

λ ∈ M with

b(w, λ) = (v, w) for all w. ( 4 . 11 )

By the definition of the functional g ,wehave g = v , and (4.10) implies

v = g = B λ ≥ β λ . Now substituting w = v in (4.11), we get

b(v, µ)
µ

b(v, λ)
λ

(v, v)
λ

sup
µ ∈ M

≥ = ≥ β v .

Thus B : V ⊥ −→ M satisfies the three conditions of Theorem 3.6, and the
mapping is an isomorphism.
Suppose condition (ii) is satisfied, i.e., B : V ⊥ −→ M is an isomorphism.
For given µ ∈ M , we determine the norm via duality:

g, µ
g

Bv,µ
Bv

µ = =

sup
g ∈ M

sup
v ∈ V ⊥

b(v, µ)
Bv

b(v, µ)
β v

= sup

v

≤ sup

v

.

V ⊥ V ⊥

∈ ∈

But then condition (i) is satisfied, and everything is proved.

Another condition which is equivalent to the inf-sup condition can be found
in Problem 4.16, where we also interpret the condition 4.2(ii) as a decomposition
property.
After these preparations, we are ready for the main theorem for saddle point
problems [Brezzi 1974]. The condition (ii) in the theorem is often referred to as the
Brezzi condition . The inf-sup condition is also called the Ladyzhenskaya-Babuska-
Brezzi condition (LBB-condition) since Ladyzhenskaya provided an inequality for
the divergence operator that is equivalent to the inf-sup condition for the Stokes
problem. Recall that as in (4.6), the kernel of B is denoted by V .

4.3 Theorem. (Brezzi's splitting theorem) For the saddle point problem (4.4), the
mapping (4.5) defines an isomorphism L : X

X × M if and only if the

× M −→

following two conditions are satisfied:
(i) The bilinear form a( · , · ) is V -elliptic, i.e.,

2

a(v, v) ≥ α v for all v ∈ V,

where α> 0 , and V is as in (4.6).
(ii) The bilinear form b( · , · ) satisfies the inf-sup condition (4.8).