Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 135

finite element spaces satisfy both conditions can often be difficult, and requires
that the finite element spaces X h and M h fit together. Practical experience shows
that enforcing these conditions is of the utmost importance for the stability of the
finite element computation.
It is useful to introduce the following notation which is analogous to (4.6):

V h : ={ v ∈ X h ; b(v, µ) = 0 for all µ ∈ M h } .

4.4 Definition. A family of finite element spaces X h ,M h is said to satisfy the
Babuska-Brezzi condition provided there exist constants α> 0 and β> 0 inde-
pendent of h such that
(i) The bilinear form a( · , · ) is V h -elliptic with ellipticity constant α> 0.
(ii)

b(v, λ h )
v

sup
v

≥ β λ h for all λ h ∈ M h . ( 4 . 16 )

∈ X h

The terminology in the literature varies. Often the condition (ii) alone is
called the Brezzi condition , the Ladyzhenskaja-Babuska-Brezzi condition ,orfor
short the LBB condition . This condition is the more important of the two, and we
will usually call it the inf-sup condition .
It is clear that - possibly after a reduction in α and β - we can take the same
constants in 4.3 and 4.4.
The following result is an immediate consequence of Lemma 3.7 and Theo-
rem 4.3.

4.5 Theorem. Suppose the hypotheses of Theorem 4.3 hold, and suppose X h ,M h
satisfy the Babuska-Brezzi conditions. Then
u − u h + λ − λ h ≤ c inf

.

v h ∈ X h u − v h + µ h ∈ M h λ − µ h

inf ( 4 . 17 )

In general, V h ⊂ V . We get a better result in the special case of conforming
approximation where V h ⊂ V . We note that in this case also the finite element
approximation of V(g) may be nonconforming for g = 0.

4.6 Definition. The spaces X h ⊂ X and M h ⊂ M satisfy condition (C) provided
V h ⊂ V , i.e., if for every v h ∈ X h , b(v h ,µ h ) = 0 for all µ h ∈ M h implies

b(v h ,µ) = 0 for all µ ∈ M .

4.7 Theorem. Suppose the hypotheses of Theorem 4.5 are satisfied along with the
condition (C). Then the solution of Problem (S h ) satisfies

u − u h ≤ c inf

v h ∈ X h u − v h .