˖ڍመڙጲ
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partition τ
h
is assumed to be uniformly regular as h tending to 0. The finite element pair
(X
h
, M
h
) is assumed to satisfy the following inf-sup condition
inf
q
h
∈M
h
sup
v
h
∈X
h
(∇ · v
h
, q
h
)
k∇v
h
k
0
≥ βkq
h
k
0
, (3)
where β is a constant independent of h.
For any nonnegative integer l, we denote by P
l
(K) the space of polynomials of degree less
than or equal to l on K. We shall denote by (X
h
, M
h
) the so-called Taylor-Hood element where
X
h
= {v
h
∈ C(Ω)
d
∩ X; v
h
|
K
∈ P
2
(K)}, M
h
= {q
h
∈ M; q
h
|
K
∈ P
1
(K)}.
Let P
h
: L
2
(Ω)
d
→ X
h
be the orthogonal projection defined by
(v − P
h
v, v
h
) = 0, ∀v ∈ L
2
(Ω)
d
, ∀v
h
∈ X
h
.
Furthermore, the projection P
h
satisfies the following properties (see [9, 23]).
kP
h
vk
0
≤ ckvk
0
, ∀v ∈ X,
kv − P
h
vk
0
+ hkv − P
h
vk
1
≤ ch
2
kAvk
0
, ∀v ∈ D(A).
(4)
We define a discrete Laplace operator A
h
: X
h
→ X
0
h
such that A
h
= −P
h
∆
h
and through
the condition that (−∆
h
u
h
, v
h
) = (∇u
h
, ∇v
h
), ∀u
h
, v
h
∈ X
h
. Thanks to [24], we know the
restriction of A
h
to X
h
is invertible, with inverse denoted by A
−1
h
. Since A
−1
h
is self-adjoint and
positive definite, we can define discrete Sobolev norms on X
h
of any order r ∈ R
kv
h
k
r
= kA
r/2
h
v
h
k
0
, ∀v
h
∈ X
h
.
These discrete norms are assumed to have various properties similar to their continuous coun-
terparts. In particular,
kv
h
k
0
= kv
h
k
L
2
, kv
h
k
1
= k∇v
h
k
0
, kv
h
k
2
= kA
h
v
h
k
0
, ∀v
h
∈ X
h
.
We introduce the discrete divergence operator B
h
: X
h
→ M
h
and discrete gradient operator
B
T
h
: M
h
→ X
0
h
such that
(B
h
v
h
, p
h
) = −(÷v
h
, p
h
) = (v
h
, B
T
h
p
h
), ∀(v
h
, p
h
) ∈ (X
h
, M
h
).
We define Y
h
= X
h
+ ∇M
h
a finite dimensional subspace of L
2
(Ω)
d
and endow Y
h
with the
norm of L
2
(Ω)
d
. We denote i
h
: X
h
→ Y
h
the natural injection and i
T
h
: Y
h
→ X
h
the usual L
2
projection. We define another version of the discrete divergence operator C
h
: Y
h
→ M
h
and
discrete gradient operator C
T
h
: M
h
→ Y
0
h
. C
h
is an extension of B
h
, i.e. C
h
i
h
= B
h
, i
T
h
C
T
h
= B
T
h
.
C
T
h
is the restriction of ∇ to M
h
, i.e. C
T
h
q
h
= ∇q
h
, ∀q
h
∈ M
h
. Due to the inf-sup condition
(3), B
h
is surjective. As a consequence, C
h
is also surjective and C
T
h
is injective. More details
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