FEM – p.1/13
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Vector Curl and Gradient, Lecture Notes - Mathematics - 10, Study notes of Mathematics
Prior error estimations
Typology: Study notes
2010/2011
1 / 13
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Vector Curl and Gradient, Lecture Notes - Mathematics - 10 and more Study notes Mathematics in PDF only on Docsity!
A prior error estimation The best approximation condition for the Galerkin finiteelement solution
u h ‖∇
u
u h
u
v h
for all
v h
S
E
in the energy norm implies that we obtain an upper boundon the error in the energy norm by evaluating or bounding ‖∇
u
v h
for
any
particular
v h
S
E
In particular taking
v h
as the interpolant of
u
from the
approximation space on each element for an arbitraryfunction
u
∈ H
1 E
generally gives a useful
a priori
error
bound.
For
a, b
, we have
H
1
C
0
so given any
u
∈ H
1 E
we can define the interpolant
v h
S
E
by
v h
node
j
u
node
j
for all
nodes
j
Hence given a partition (a mesh) a
x 0 < x 1 < x 2 <... < x n
b ‖∇
u
u h
2
u
v h
2
n ∑ j =
x j x j − 1
u ′
x
v ′ h
x
2 d x
and we only have to find out how closely a continuousfunction can be approximated by the polynomial interpolantof the appropriate degree.Write
e I
u
v h
for the error in interpolation of
u
by
v h FEM – p.4/
eg. if
P 1
approximation is used
j−
x
x
x
j^ j+
x
j+
x
j−
and
P 2
approximation on a single element looks like
With
h j
x j
x j − 1
we have for
s
x j − 1 , x j
e ′ I
s
s ξ e ′′ I
t ) d t
so
[
e ′ I
s
)]
2
[∫
s ξ
e ′′ I
t ) d t
]
2 ≤
s ξ
2 d t
s ξ
[
e ′′ I
t
)]
2 d t
(C-S)
h j
x j x j − 1
[
u ′′
t
)]
2 d t
as
e ′′ I
u ′′
v ′′ h
u ′′
because
v h
is linear on
x j − 1 , x j
. FEM – p.7/
[
e ′ I
s
)]
2
h j
x j x j − 1
[
u ′′
t
)]
2 d t
Thus integrating
x j x j − 1
[
e ′ I
s
)]
2 d s
x j x j − 1 h j
x j x j − 1
[
u ′′
t
)]
2 d t d s = h 2 j
x j x j − 1
[
u ′′
t
)]
2 d t
and summing over
j
,... , n
writing
h = max j
h j
b a
[
e ′ I
s
)]
2 d s
h 2
b a
[
u ′′
t
)]
2 d t
which on taking square roots is
e ′ I
h
u ′′
h
f
By a similar arguement (see exercises) we can show that
e I
h
e ′ I
for any polynomial interpolant
v h
of
u
on such a partition
and hence in the case of
P 1
approximation we have
e I
h
e ′ I
h 2
u ′′
h 2
f
This does not yet give us a convergence bound for theGalerkin finite element solution in the
L
2
a, b
norm which
we will come to later.
Employing
P 2
approximation on
a, b
, we let
e I
u
v h
where
v h
interpolates
u
at each
x j
and also at
each midpoint
1 2
x j
x j − 1
. So if we assume that
u ′′′
exists then applying the Mean Value Theorem twice on anyinterval
x j − 1 , x j
we see that there exists
ξ
x j − 1 , x j
with
e ′′ I
ξ
Arguement similar to the above then gives
b a
[
e ′′ I
s
)]
2 d s
h 2
b a
[
u ′′′
t
)]
2 d t
and further
b a
[
e ′ I
s
)]
2 d s
h 2
b a
[
e ′′ I
s
)]
2 d s ≤ h 4
b a
[
u ′′′
t
)]
2 d t