Lecture 6
Notes on Finite Element Methods
Lecture 6
Integration error for Lagrange finite elements
§1 Bramble-Hilbert lemma
We will first prove the Sobolev lemma.
Theorem 6.1 (Sobolev lemma). Let Ω be a finite union of star-shaped domains.
Then W m,p (Ω) ,→ C(Ω) if m > np , m = 1 is allowed if p = 1.
Proof. Let u ∈ W m,p (Ω). Then there exists a sequence {uk } ⊂ C ∞ (Ω) such that
uk → u in W m,p .
kuj − uk k∞ ≤ kQm (uj − uk )k∞ + kRm (uj − uk )k∞
≤ Ckuj − uk kL1 + C|uj − uk |W m,p → 0 as k, j → ∞.
It therefore follows that {uj } is Cauchy in L∞ and there exists w ∈ C(Ω) such
that uk → w in L∞ ; but uj → u in L1 so u = w almost everywhere.
We claim that ∂ β Qm u = Qm−|β| ∂ β u. To see this look at derivatives of
Tym u(x) =
X (x − y)α
∂ α u(y).
α!
|α|<m
We have ∂xβ (x − y)α =
∂ β Tym u(x) =
α!
(x
(α−β)!
X
|α|<m, α≥β
− y)α−β ,
(x − y)α−β α−β β
∂
∂ u(y) =
(α − β)!
X
|γ|<m−|β|
(x − y)γ γ β
∂ ∂ u(y).
γ!
We now prove Young’s inequality using duality.
ZZ
u(x − y)v(y)ϕ(x) dydx
hu ∗ v, ϕi =
ZZ
=
ũ(y − x)ϕ(x)v(y) dxdy
= hũ ∗ ϕ, vi ,
provided that h|ũ ∗ ϕ, |v|i < ∞. Note that ũ = u(−x).
h|ũ ∗ ϕ|, |v|i ≤ kũ ∗ ϕkL∞ kvkL1 ≤ kukLp kϕkLq kvkL∞ ,
which makes | hu ∗ v, ϕi | ≤ kukp kϕkq kvk∞ . By duality
ku ∗ vkLp ≤ sup
ϕ∈Lq
hu ∗ v, ϕi
≤ kukLp kvkL1 .
kϕkLq
25
(6.1)
Lecture 6
Notes on Finite Element Methods
Remark. Young’s inequality says for 1 +
1
r
=
1
p
+ 1q , ku ∗ vkr ≤ kukp kvkq .
Theorem 6.2 (Bramble-Hilbert lemma). Let u ∈ W m,p (Ω) with 0 ≤ k ≤ m.
Then
(6.2)
inf ku − vkW k,p (Ω) ≤ C|h|m−k |u|W m,p (Ω) .
v∈Pm−1
Proof. Let u ∈ C ∞ (Ω). Estimate (5.11) gives
kRm ukLp ≤ CkIm,h kL1 |u|W m,p ≤ Chm |u|W m,p ,
and
∂ β Rm u = ∂ β (u − Qm u) = ∂ β u − Qm−|β| ∂ β u = Rm−|β| ∂ β u.
We conclude for smooth u
|Rm u|W k,p ≤ max kRm−k ∂ β ukLp ≤ Chm−k |u|W m,p .
|β|=k
Let now {uk } ⊂ C ∞ (Ω) be a sequence with un → u in W m,p . By Lemma 5.3
kQm uj − Qm uν kW k,p ≤ Ckuj − uν kL1 → 0,
so define Qm u = limj→∞ Qm uj ,
|u − Qm u|W k,p ≤ |u − uj |W k,p + |uj − Qm uj |k,p + |Qm uj − Qm u|W k,p .
By assumption, |u − uj |k,p , |Qm uj − Qm u|k,p → 0 and by the above argument for
smooth functions,
|u − Qm u|W k,p ≤ C 0 hm−k |uj |W m,p + o(1) ≤ Chm−k |u|W m,p + o(1),
as j → ∞.
§2 Interpolation error for the Lagrange finite element
Let Pd−1 be the shape function space defined on a finite element domain τ , τ
being an n-simplex.
The standard simplex in Rn is given
(
σ=
x ∈ Rn−1 :
n−1
X
)
xi = 1, xi ≥ 0 .
(6.3)
i=1
where its vertices are (conveniently) taken to be spanned by {0, e1 , ..., en }, ei
being the standard unit vector in Rn . The dimension of Pd−1 = #{σ ∩ Zn }.
There exists an affine bijection A : σ → τ . Take
26
Lecture 6
Notes on Finite Element Methods
N = {δz : z ∈ A(σ ∩ Zn )},
evidently, #N = dim Pd−1 . Let Vτ denote A(σ ∩ Zn ), the set of points which
characterizes the shape functions; these determine the degrees of freedom. Recalling Definition 4.4, we have an equivalent definition for the local interpolation
for a function u,
X
Iτ u =
u(z)φz ,
z∈Vτ
defined for u ∈ W m,p with m > n/p. We wish to determine an estimate on
ku − Iτ ukW k,p (τ ) :
ku − Iτ ukk,p ≤ ku − Qd ukk,p + kQd u − Iτ ukk,p .
Noting that ku − Qd ukk,p ≤ Chd−k |u|d,p by the Bamble-Hilbert lemma, we now
study the behaviour of the second term. We first estimate kIτ kW k,p (τ ) :
X
kIτ vkk,p ≤
kφz kk,p kvk∞ .
z∈Vτ
Take h = diam τ , and consider a function φb whose domain τ is modified so
b
that the modification is of size R1; i.e. φz (x) = φ(x/h).
From this, we conclude
k
−k
k
p
n−kp
that D φz ∼ h which makes τ |D φz | ∼ h
hence kφz kk,p ≤ Chn/p−k . It
therefore follows that
n
kIτ vkk,p ≤ Ch−k+ p kvk∞ ,
v ∈ W m,p (τ ).
Noting that Iτ Qd u = Qd u, we conclude that
n
kQd u − Iτ ukk,p = kIτ (u − Qd u)kk,p ≤ Ch−k+ p ku − Qd uk∞ .
Figure 3: An example of a standard simplex in two and three dimensions.
27
Lecture 6
Notes on Finite Element Methods
We wish to bound the the term
ku − Qd uk∞ , present in the last expression, we
n
recall that kRm uk∞ ≤ Chm− p |u|m,p , 0 ≤ k ≤ d ≤ m with d corresponding to the
order of the finite element space and m > n/p is chosen so as to ensure u ∈ C.
Here, we do not wish to restrict ourself to finite element spaces of dimensions
satisfying such restrictions as d > n/p so as to ensure necessary regularity. For
this, we consider the following argument.
n
ku − Qd uk∞ ≤ ku − Qm uk∞ + kQm u − Qd uk∞ ≤ Chm− p |u|m,p + kQm u − Qd uk∞ ,
but
Qm u(x) − Qd u(x) =
X Z ∂ α u(y)
(x − y)α φ(y) dy,
α!
B
d≤|α|<m
therefore
kQm u − Qd uk∞ ≤ C
m−1
X
hj |u|j,p kφkLq ≤ C
m−1
X
n
hj+ q −n |u|j,p .
j=d
j=d
We have
ku − Iτ ukW k,p (τ ) ≤ C
m
X
n
n
hj− p −k+ p |u|j,p
j=d
= C hd−k |u|d,p + · · · + hm−k |u|m,p ≤ Chd−k kukW m,p (τ ) .
Remark. The previous implies that the accuracy/rate of convergence depends
on the order of polynomial used.
Since
ku − IP ukpW k,p (Ω) =
X
ku − Iτ ukpW k,p (τ ) ,
(6.4)
τ ∈P
we conclude
inf
v∈S d (P )
ku − vkW k,p (Ω) ≤ Chd−k kukW m,p (Ω) ,
where h = maxτ ∈P diam τ .
28
(6.5)