Civil Engineering Reference
In-Depth Information
Note that the condition in Fortin's criterion can be checked without referring
explicitly to the norm of the Lagrange multipliers. This is an advantage when the
space of the Lagrange multipliers is equipped with an exotic norm, and it is thus
used for example when the Lagrange multipliers belong to trace spaces.
4.9 Remark.
There is a converse statement to Fortin's criterion.
If the finite
element spaces X
h
, M
h
satisfy the inf-sup condition, then there exists a bounded
linear projector
h
:
X
→
X
h
such that (4.18) holds.
Indeed, given
v
∈
X
, define
u
h
∈
X
h
as the solution of the equations
(u
h
,w)
+
b(w, λ
h
)
=
(v, w)
for all
w
∈
X
h
,
(
4
.
19
)
b(u
h
,µ)
=
b(v, µ)
for all
µ
∈
M
h
.
Since the inner product in
X
is coercive by definition, the problem is stable, and
from Theorem 4.3 it follows that
u
h
≤
c
v
.
Moreover, a linear mapping is defined by
v
−→
v
:
=
u
h
, and the proof is
complete.
The linear process defined above is called
Fortin interpolation.
As a corollary we obtain a relationship between the approximation with the
constraint induced by the bilinear form
b
and the approximation in the larger finite
element space
X
h
.
4.10 Remark.
If the spaces
X
h
and
M
h
satisfy the inf-sup condition, then there
exists a constant
c
independent of
h
such that for every
u
∈
V(g)
,
v
h
∈
V
h
(g)
u
−
v
h
≤
c
inf
inf
w
h
∈
X
h
u
−
w
h
.
Proof.
We make use of Fortin interpolation. Obviously,
h
w
h
=
w
h
for each
w
h
∈
X
h
.Given
u
∈
V(g)
we have
h
u
∈
V
h
(g)
and
u
−
h
u
=
u
−
w
h
−
h
(u
−
w
h
)
≤
(
1
+
c)
u
−
w
h
.
Since this holds for all
w
h
∈
X
h
, the proof is complete.
Sometimes error estimates are wanted for some norms for which not all
hypotheses of Theorem 4.3 hold. In this context we note that the norm of the
bilinear form
b
does not enter into the a priori estimate (4.12). This fact is used
for the estimate of
λ
−
λ
h
when an estimate of
u
−
u
h
has been established
by applying other tools.
Search WWH ::
Custom Search