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Error Bounds and Approximation in Finite Element Method (FEM), Study notes of Mathematics
The construction of error bounds for 2-dimensional problems in finite element method (fem) using the best approximation and coercivity and continuity of a bilinear form. It also covers the lax-milgram lemma and cea's lemma, which provide existence and uniqueness of solutions and error bounds for the galerkin finite element method.
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2010/2011
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The construction of error bounds for 2-dimensionalproblems is similar though more technical:The assumption that
u
∈ H
2
allows one to use the best
approximation
u
u
h
u
v
h
for all
v
h
S
E
in the energy norm to obtain a bound by considering theinterpolant
v
h
S
E
Higher degree piecewise polynomial elements as in1-dimension give better accuracy (faster convergence) forsmooth enough solutions.Similar results apply for
Q
k
approximations.
So far we have considered only the weak form for Poissonproblems for which we have been able to show existenceand uniqueness in the appropriate functions spaces.For such problems the Galerkin finite element solution hasbeen shown to give the best possible approximation from S
E
to the weak solution in the energy norm. However
existence and uniqueness apply to a much greater set ofproblems and near-best approximation results for thecorresponding Galerkin approximations hold also for manyother problems.
We require a little more mathematical machinery to includesuch problemsDefinition: A real
Hilbert Space
V
is a vector space with an
inner product
V
V
×
V
R
and associated norm
V
V
in which all Cauchy
sequences converge to limit points in
V
Examples:
V
L
2
v, w
V
v, w
Ω
vw
V
H
1
v, w
V
Ω
vw
Ω
v
w
(this is the
H
1
norm)
V
H
1 E
0
v, w
V
a
v, w
Ω
v
w
Examples:
Theorem (Cea’s Lemma)Suppose that
V
is a real Hilbert space and that the bilinear
form
a
and the linear functional
`
satisfy
a
u, u
γ
u
2 V
for all
u
V
a
u, v
u
V
v
V
for all
u, v
V
`
v
v
V
for all
v
V.
If
S
V
is a vector subspace with
u
h
S
satisfying
a
u
h
, v
h
`
v
h
for all
v
h
S
then
u
u
h
V
Γ^ γ
u
v
h
V
for all
v
h
S
Proof: For any
v
h
S
, subtracting
a
u
h
, v
h
`
v
h
from
a
u, v
h
`
v
h
gives
a
u
u
h
, v
h
for all
v
h
S
So
a
u
u
h
, u
u
h
a
u
u
h
, u
v
h
a
u
u
h
, v
h
u
h
a
u
u
h
, u
v
h
and we have
γ
u
u
h
2 V
a
u
u
h
, u
u
h
a
u
u
h
, u
v
h
u
u
h
V
u
v
h
V
so that
u
u
h
V
/γ
u
v
h
V
FEM – p.14/
The rather neat
Aubin-Nitsche
duality argument allows
bounds on the Galerkin finite element error in the
L
2
norm which are found to be tight in computations:Suppose that
a
is coercive and continuous and
`
continuous in
V
and
S
V
is a vector subspace.
is
the
L
2
inner product
(W1):
a
u, v
`
v
for all
v
V
(G1):
a
u
h
, v
h
`
v
h
for all
v
h
S
V
Let
e
u
u
h
u
u
h
and define
w, w
h
by
(W2):
a
v, w
e, v
for all
v
V
(G2):
a
v
h
, w
h
e, v
h
for all
v
h
S
V
u
u
h
e, u
u
h
a
u
u
h
, w
(W2)
a
u
u
h
, w
a
u
u
h
, w
h
(Galerkin orthogonali
a
u
u
h
, w
w
h
u
u
h
V
w
w
h
V
(Continuity)
3
γ
2
u
v
h
V
w
z
h
V
for all
v
h
, z
h
S
(Cea)
That is ‖
u
u
h
L
2
(Ω)
3
γ
2
u
v
h
V
w
z
h
V
for all
v
h
, z
h
S
FEM – p.17/
Recapitulating: for the Galerkin finite element error with
P
1
elements we have
u
u
h
C
h
in the energy norm, and Aubin-Nitsche gives us
u
u
h
‖ ≤ C
h
2
for some constants
C
C
For
P
2
elements correspondingly:
u
u
h
C
h
2
in the energy norm, and Aubin-Nitsche gives us
u
u
h
C
h
4