Civil Engineering Reference
In-Depth Information
§ 5. Mixed Methods for the Poisson Equation
The treatment of the Poisson equation by mixed methods already elucidates some
characteristic features and shows that saddle point formulations are not only useful
for minimization problems with given constraints as in (4.1), (4.2). For example,
there are two different pairs of spaces for which the saddle point problem is stable in
the sense of Babuska and Brezzi. It is interesting that different boundary conditions
turn out to be natural conditions in the two cases.
The method, often called the
dual mixed method
, has been well established
for a long time. On the other hand, the
primal mixed method
has recently attracted
a lot of interest since it shows that mixed methods are often related to a softening
of the energy functional and how elasticity problems with a small parameter can
be treated in a robust way.
Moreover there are special results if
X
or
M
coincides with an
L
2
-space.
The Poisson Equation as a Mixed Problem
The Laplace equation or the Poisson equation
u
=
div grad
u
=−
f
can be
written formally as the system
grad
u
σ,
div
σ
=−
f.
=
(
5
.
1
)
d
. Then (5.1) leads directly to the following saddle point problem: Find
(σ, u)
∈
L
2
()
d
Let
⊂ R
×
H
0
()
such that
for all
τ
∈
L
2
()
d
,
(σ, τ )
0
,
−
(τ,
∇
u)
0
,
=
0
(
5
.
2
)
for all
v
∈
H
0
().
−
(σ,
∇
v)
0
,
=−
(f, v)
0
,
These equations can be treated in the general framework of saddle point problems
with
=
L
2
()
d
,M
:
=
H
0
(),
X
:
(
5
.
3
)
a(σ, τ)
:
=
(σ, τ )
0
,
,
b(τ, v)
:
=−
(τ,
∇
v)
0
,
.
The linear forms are continuous, and
a
is obviously
L
2
-elliptic. To check the inf-
sup condition, we use Friedrichs' inequality in a similar way as for the original
minimum problem in Ch. II, §2. Given
v
∈
H
0
()
, consider the quotient appearing
in the condition for
τ
:
=−∇
v
∈
L
2
()
d
:
τ
0
=
−
(τ,
∇
v)
0
,
(
∇
v,
∇
v)
0
,
∇
v
0
b(τ, v)
1
c
v
1
.
=
=|
v
|
1
≥
τ
0
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