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C
4
(J
∈
×
Theorem 2.3.8
If u
G)
,
we have
1
≤
θ<
2
(i)
Fo r
2
<θ
≤
1
or for
0
and
(
2.20
),
h
2
ε
m
C(u)(h
2
su
m
2
≤
+
k)
;
1
(ii)
Fo r θ
=
2
,
h
2
ε
m
C(u)(h
2
k
2
),
su
m
2
≤
+
where the constant C(u) >
0
depends on the exact solution u and its derivatives
.
We next explain the finite element method which is based on variational formu-
lations of the differential equations.
2.3.3 Finite Element Method
For the discretization with finite elements, we use the
method of lines
where we first
only discretize in space to obtain a system of coupled ordinary differential equations
(ODEs). In a second step, a time discretization scheme is applied to solve the ODEs.
We do not require the PDE (
2.4
) to hold pointwise in space but only in the varia-
tional sense. Therefore, we fix
t
C
0
∈
J
and let
v
∈
(G)
be a smooth test function
satisfying
v(a)
0. We multiply the PDE with
v
, integrate with respect to
the space variable
x
and use integration by parts to obtain
=
v(b)
=
b
b
b
∂
t
uv
d
x
−
∂
xx
uv
d
x
=
fv
d
x,
a
a
a
∂
x
u(x,t)v(x)
x
=
b
x
=
a
b
b
b
d
d
t
⇒
uv
d
x
−
+
∂
x
u∂
x
v
d
x
=
fv
d
x.
a
a
a
=
0
Therefore, we have
b
b
b
d
d
t
C
0
uv
d
x
+
∂
x
u∂
x
v
d
x
=
fv
d
x,
∀
v
∈
(G).
(2.21)
a
a
a
Since
C
0
(G)
is not a closed subspace of
L
2
(G)
, we will consider test functions in
the
Sobolev space H
0
(G)
which is the closure of
C
0
(G)
in the
H
1
-norm,
2
2
u
2
u
H
1
(G)
=
u
L
2
(G)
+
L
2
(G)
.
The Sobolev space
H
0
(G)
consists of all continuous functions which are piecewise
differentiable and vanish at the boundary. Since (
2.21
) holds for all
v
C
0
∈
(G)
,
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