Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 159

By Remark 6.1, the first equation in (6.5) can be written in the form

grad p, v) 0 , for all v ∈ H 0 () n .
Since u ∈ C 2 () n , by the theory of scalar equations in Ch. II, §2, it follows that
u is a classical solution of
−

( grad u, grad v) 0 , = (f −

u = f − grad p in ,

u = 0 on ∂,

and the proof is complete.

The inf-sup Condition

In order to apply the general theory described in the previous section, let

V : ={ v ∈ X ; ( div v, q) 0 , = 0 for all q ∈ L 2 () } .

grad u 0 , = a(u, u) 1 / 2 is a norm on X .
Hence, the bilinear form a is H 0 -elliptic. Thus it is elliptic not only on the subspace
V , but also on the entire space X . This means that we could get by with an even
simpler theory than in §4.
In order to ensure the existence and uniqueness of a solution of the Stokes
problem, it remains to verify the Brezzi condition.
By the abstract Lemma 4.2, the inf-sup condition can be expressed in terms
of properties of the operators B and B . In the concrete case of the Stokes equation
with b(v, q) = ( div v, q) 0 , =− (v, grad q) 0 , , the conditions are to be under-
stood as properties of the operators div and grad. They are presented in the next
two theorems. Their proof is beyond the scope of this topic; cf. Duvaut and Lions
[1976].
The following result on the divergence is attributed to Ladysenskaya. Recall

By Friedrichs' inequality, | u | 1 , =

that

V ⊥ :

={ u ∈ X ; ( grad u, grad v) 0 , = 0 for all v ∈ V } ( 6 . 7 )

is the H 1 -orthogonal complement of V .

n be a bounded connected domain with Lipschitz contin-
uous boundary. Then the mapping
div : V ⊥ −→

6.3 Theorem. Let ⊂ R

L 2 , 0 ()

div v
is an isomorphism. Moreover, for any q ∈ L 2 () with qdx =

v −→

0 , there exists a

function v ∈ V ⊥ ⊂ H 0 () n

with

div v = q and v 1 , ≤ c q 0 , , ( 6 . 8 )

where c = c() is a constant.