C0 Interior Penalty Methods
Error Analysis
Current Research in Finite Element Methods
CIMPA Summer School
Mumbai, July 2015
Outline
I
Elliptic Regularity
I
Standard A Priori Analysis
I
Medius Error Analysis
I
A Posteriori Error Analysis
Elliptic Regularity
Biharmonic Equation
∆2 u = f
in Ω
with different boundary conditions
Ω = bounded polygonal domain in R2
f ∈ L2 (Ω)
Biharmonic Equation
∆2 u = f
in Ω
with different boundary conditions
Ω = bounded polygonal domain in R2
f ∈ L2 (Ω)
Variational/Weak Formulation Find u ∈ V such that
Z
Z
D2 u : D2 v dx =
f v dx
∀v ∈ V
Ω
Ω
Biharmonic Equation
∆2 u = f
in Ω
with different boundary conditions
Ω = bounded polygonal domain in R2
f ∈ L2 (Ω)
Variational/Weak Formulation Find u ∈ V such that
Z
Z
D2 u : D2 v dx =
f v dx
∀v ∈ V
Ω
Ω
Boundary Conditions of Clamped Plates
V = H02 (Ω)
(u = ∂u/∂n = 0 on ∂Ω)
Biharmonic Equation
∆2 u = f
in Ω
with different boundary conditions
Ω = bounded polygonal domain in R2
f ∈ L2 (Ω)
Variational/Weak Formulation Find u ∈ V such that
Z
Z
D2 u : D2 v dx =
f v dx
∀v ∈ V
Ω
Ω
Boundary Conditions of Simply Supported Plates
V = H 2 (Ω) ∩ H01 (Ω)
(u = ∆u = 0 on ∂Ω)
Biharmonic Equation
∆2 u = f
in Ω
with different boundary conditions
Ω = bounded polygonal domain in R2
f ∈ L2 (Ω)
Variational/Weak Formulation Find u ∈ V such that
Z
Z
D2 u : D2 v dx =
f v dx
∀v ∈ V
Ω
Ω
Boundary Conditions of the Cahn-Hilliard Type
V = {v ∈ H 2 (Ω) : ∂v/∂n = 0 on ∂Ω}
(∂u/∂n = ∂(∆u)/∂n = 0 on ∂Ω)
Shift Theorem
If Ω is a smooth domain, then f ∈ L2 (Ω) implies the solution u
of the variational/weak problem belongs to H 4 (Ω).
Agmon-Douglis-Nirenberg (1959)
Shift Theorem
If Ω is a smooth domain, then f ∈ L2 (Ω) implies the solution u
of the variational/weak problem belongs to H 4 (Ω).
Agmon-Douglis-Nirenberg (1959)
When Ω is a polygonal domain, u is still an H 4 function away
from the corners of Ω, but u does not belong to H 4 (Ω) in general even if f ∈ C∞ (Ω).
Shift Theorem
If Ω is a smooth domain, then f ∈ L2 (Ω) implies the solution u
of the variational/weak problem belongs to H 4 (Ω).
Agmon-Douglis-Nirenberg (1959)
When Ω is a polygonal domain, u is still an H 4 function away
from the corners of Ω, but u does not belong to H 4 (Ω) in general even if f ∈ C∞ (Ω).
There exists α ≤ 2, depending on the interior angles of Ω and
the boundary conditions, such that the solution u of the model
problem belongs to H 2+α (Ω) when f ∈ L2 (Ω) and we have
kukH 2+α (Ω) ≤ CΩ,α kf kL2 (Ω)
Shift Theorem
If Ω is a smooth domain, then f ∈ L2 (Ω) implies the solution u
of the variational/weak problem belongs to H 4 (Ω).
Agmon-Douglis-Nirenberg (1959)
When Ω is a polygonal domain, u is still an H 4 function away
from the corners of Ω, but u does not belong to H 4 (Ω) in general even if f ∈ C∞ (Ω).
There exists α ≤ 2, depending on the interior angles of Ω and
the boundary conditions, such that the solution u of the model
problem belongs to H 2+α (Ω) when f ∈ L2 (Ω) and we have
kukH 2+α (Ω) ≤ CΩ,α kf kL2 (Ω)
α = index of elliptic regularity
Shift Theorem
Fractional Order Sobolev Space H 2+s (Ω)
(0 < s < 1)
A function v belongs to H 2+s (Ω) if and only if
v belongs to H 2 (Ω)
if w is a second order (weak) derivative of v, then
Z Z
|w(x) − w(y)|2
dx dy < ∞
|x − y|2+2s
Ω Ω
Shift Theorem
Fractional Order Sobolev Space H 2+s (Ω)
(0 < s < 1)
A function v belongs to H 2+s (Ω) if and only if
v belongs to H 2 (Ω)
if w is a second order (weak) derivative of v, then
Z Z
|w(x) − w(y)|2
dx dy < ∞
|x − y|2+2s
Ω Ω
The norm of H 2+s (Ω) is defined by
kvk2H 2+s (Ω)
=
kvk2H 2 (Ω)
X Z Z |(∂ β v)(x) − (∂ β v)(y)|2
+
dx dy
|x − y|2+2s
Ω Ω
|β|=2
Shift Theorem
Boundary Conditions of Clamped Plates
u=
∂u
=0
∂n
on ∂Ω
The index of elliptic regularity α satisfies
1
<α≤2
2
α > 1 if Ω is convex.
α < 1 if Ω is non-convex.
Shift Theorem
Boundary Conditions of Simply Supported Plates
u = ∆u = 0
on ∂Ω
The index of elliptic regularity α satisfies
0<α≤2
For a rectangle, α = 2.
For an L-shaped domain, α can be any number < 13 .
α can be close to 0 if there is an interior angle of Ω close to π.
(This can happen for a convex Ω.)
Shift Theorem
Boundary Conditions of Cahn-Hilliard Type
∂u/∂n = ∂(∆u)/∂n = 0
on ∂Ω
The index of elliptic regularity α satisfies
0<α≤2
For a rectangle, α = 2.
For an L-shaped domain, α can be any number < 13 .
α can be close to 0 if there is an interior angle of Ω close to π.
(This can happen for a convex Ω.)
A Second Order Model Problem
Find u ∈ H01 (Ω) such that
Z
Z
∇u · ∇v dx =
fv dx
Ω
Ω
∀ v ∈ H01 (Ω)
where Ω ⊂ R2 is a polygonal domain and f ∈ L2 (Ω).
A Second Order Model Problem
Find u ∈ H01 (Ω) such that
Z
Z
∇u · ∇v dx =
fv dx
Ω
Ω
∀ v ∈ H01 (Ω)
where Ω ⊂ R2 is a polygonal domain and f ∈ L2 (Ω).
Shift Theorem
If Ω is smooth, then u belongs to H 2 (Ω).
A Second Order Model Problem
Find u ∈ H01 (Ω) such that
Z
Z
∇u · ∇v dx =
fv dx
Ω
Ω
∀ v ∈ H01 (Ω)
where Ω ⊂ R2 is a polygonal domain and f ∈ L2 (Ω).
Shift Theorem
If Ω is smooth, then u belongs to H 2 (Ω).
If Ω is a non-convex polygonal domain, then u does not belong
to H 2 (Ω) in general.
A Second Order Model Problem
Let ω > π be the interior angle θ = 0
at an vertex p of Ω, and (r, θ) be
the polar coordinates at p.
θ=ω
p
Consider the function
Ω
ψ(r, θ) = r(π/ω) sin (π/ω)θ φ(r)
PSfrag replacements
where φ(r) is a smooth
cut-off
function that equals 1 near 0 and
vanishes for r ≥ δ > 0.
It is clear that ψ vanishes on the
two edges at p and, if δ is sufficiently small, ψ also vanishes on
the rest of ∂Ω.
ω
φ
1
r
δ
A Second Order Model Problem
Away from the vertex p the function
ψ(r, θ) = r(π/ω) sin (π/ω)θ φ(r)
is smooth up to the boundary of Ω. Near the vertex p, ψ is
square integrable and
#
"
sin (π/ω) − 1 θ
(π/ω)−1
(∇ψ)(r, θ) = (π/ω)r
cos (π/ω) − 1 θ
is also square integrable since (π/ω) − 1 >
1
2
− 1 = − 12 .
Therefore ψ belongs to H 1 (Ω) and hence ψ ∈ H01 (Ω) since ψ
vanishes on ∂Ω.
A Second Order Model Problem
ψ(r, θ) = r(π/ω) sin (π/ω)θ φ(r)
= r(π/ω) sin (π/ω)θ = Im(z(π/ω) )
(z = reiθ )
near p
A Second Order Model Problem
ψ(r, θ) = r(π/ω) sin (π/ω)θ φ(r)
= r(π/ω) sin (π/ω)θ = Im(z(π/ω) )
Therefore
−∆ψ = 0
near p
and hence
F = −∆ψ ∈ C∞ (Ω̄)
near p
A Second Order Model Problem
ψ(r, θ) = r(π/ω) sin (π/ω)θ φ(r)
= r(π/ω) sin (π/ω)θ = Im(z(π/ω) )
near p
Therefore
−∆ψ = 0
near p
and hence
F = −∆ψ ∈ C∞ (Ω̄)
ψ ∈ H01 (Ω) satisfies
Z
Z
∇ψ · ∇v dx =
Fv dx
Ω
Ω
But ψ does not belong to H 2 (Ω).
∀ v ∈ H01 (Ω)
A Second Order Model Problem
Near the corner p
ππ
∂2ψ
=
−
1
r(π/ω)−2 sin (π/ω) − 2 θ
2
ω ω
∂x1
is not square integrable because
(π/ω) − 2 < 1 − 2 = −1
We will refer to ψ as a singular function associated with the
reentrant corner.
ψ ∈ H 1+(π/ω)− (Ω)
for any > 0, but
ψ 6∈ H 1+(π/ω) (Ω)
A Second Order Model Problem
Fractional Order Sobolev Space H 1+s (Ω)
(0 < s < 1)
A function v belongs to H 1+s (Ω) if and only if
v belongs to H 1 (Ω)
if w is a first order (weak) derivative of v, then
Z Z
|w(x) − w(y)|2
dx dy < ∞
|x − y|2+2s
Ω Ω
The norm of H 1+s (Ω) is defined by
kvk2H 1+s (Ω)
=
kvk2H 2 (Ω)
X Z Z |(∂ β v)(x) − (∂ β v)(y)|2
+
dx dy
|x − y|2+2s
Ω Ω
|β|=1
A Second Order Model Problem
Singular Function Representation In general we have
X
u = uR +
cj ψj
ωj >π
where
uR ∈ H 2 (Ω) ∩ H01 (Ω)
ω1 , ω2 , . . . are the interior angles at the vertices of Ω, cj ’s are
constants, and
(π/ωj )
ψj = rj
sin (π/ωj )θj φj (rj )
is the singular function associated with the re-entrant corner
with interior angle ωj .
A Second Order Model Problem
Singular Function Representation In general we have
X
u = uR +
cj ψj
ωj >π
where uR ∈ H 2 (Ω) ∩ H01 (Ω). In particular
u = uR ∈ H 2 (Ω)
if Ω is convex, but in general
π
ω
if Ω is nonconvex, where ω is the largest re-entrant corner.
u ∈ H s (Ω)
for all s < 1 +
A Second Order Model Problem
Singular Function Representation In general we have
X
u = uR +
cj ψj
ωj >π
where uR ∈ H 2 (Ω) ∩ H01 (Ω). In particular
u = uR ∈ H 2 (Ω)
if Ω is convex, but in general
π
ω
if Ω is nonconvex, where ω is the largest re-entrant corner.
u ∈ H s (Ω)
for all s < 1 +
Moreover, we have an elliptic regularity estimate
X
|uR |H 2 (Ω) +
|cj | ≤ CΩ kf kL2 (Ω)
ωj >π
A Second Order Model Problem
Example
3π
2
for an L-shaped domain and hence the solution of the Poisson
problem in general only belongs to
ω=
H s (Ω)
for
s<
π
5
=1+
3
ω
A Second Order Model Problem
Shift Theorem Revisited
Given F ∈ H −1 (Ω) = [H01 (Ω)]0 , there is a unique u ∈ H01 (Ω) such
that
Z
∇u · ∇v dx = F(v)
∀ v ∈ H01 (Ω)
Ω
Moreover we have
kukH 1 (Ω) ≤ CΩ kFkH −1 (Ω)
where
kFkH −1 (Ω) =
sup
v∈H01 (Ω)
|F(v)|
kvkH 1 (Ω)
Therefore the shift theorem holds for H −1 (Ω). On the other
hand, if Ω is nonconvex, then the shift theorem fails for H 0 (Ω) =
L2 (Ω).
A Second Order Model Problem
Shift Theorem Revisited
Let Ω be a nonconvex polygonal domain with maximum reentrant angle ω, and u ∈ H01 (Ω) satisfy
Z
∇u · ∇v dx = F(v)
∀ v ∈ H01 (Ω)
Ω
where F ∈ H −1+s (Ω) for some s ∈ (0, π/ω). Then we have
u ∈ H 1+s (Ω) and
kukH 1+s (Ω) ≤ CΩ,s kFkH −1+s (Ω)
For 0 < t < 1, H −t (Ω) is the subspace of H −1 (Ω) = [H01 (Ω)]0
consisting of linear functionals G such that
kGkH −t (Ω) =
|G(v)|
<∞
v∈H 1 (Ω) kvkH t (Ω)
sup
0
References
• Grisvard, Elliptic Problems in Non-Smooth Domains,
SIAM, 2011
• Dauge, Elliptic Boundary Value Problems on Corner Do-
mains, Springer-Verlag, 1988
• Nazarov and Plamenevsky, Elliptic Problems in Domains
with Piecewise Smooth Boundaries, de Gruyter, 1994
• Blum and Rannacher, On the boundary value problem of
the biharmonic operator on domains with angular corners,
Math. Methods Appl. Sci., 1980.
Standard A Priori Error Analysis
Boundary Conditions of Clamped Plates
Ω = bounded polygonal domain
∆2 u = f
∂u
=0
u=
∂n
f ∈ L2 (Ω)
in Ω
on ∂Ω
Variational/Weak Formulation
Find u ∈ H02 (Ω) such that
Z
a(u, v) =
fv dx
Ω
Z
a(w, v) =
Ω
D2 w : D2 v dx
∀ v ∈ H02 (Ω)
D2 w : D2 v =
2
X
i,j=1
wxi xj vxi xj
Discrete Problem
Find uh ∈ Vh such that
Z
ah (uh , v) =
f v dx
∀ v ∈ Vh
Ω
n ∂ 2 w oohh ∂v ii
XZ n
ds
ah (w, v) =
D w : D v dx +
∂n2
∂n
T∈Th T
e∈Eh e
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
+
ds
|e| e ∂n
∂n
XZ
2
2
e∈Eh
Eh = set of edges
|e| = length of e
{{·}} = average
[[·]] = jump
σ = penalty parameter
Galerkin Orthogonality
The C0 interior penalty method is consistent in the sense that
Z
(∗)
ah (u, v) =
fv dx
∀ v ∈ Vh
Ω
This was derived earlier under the assumption that u is sufficiently smooth. Since u is not very smooth at the corners of
Ω, one must justify carefully the integration by parts involving
u around the corners. This can be done by using the singular
function representation of u.
Galerkin Orthogonality
The C0 interior penalty method is consistent in the sense that
Z
(∗)
ah (u, v) =
fv dx
∀ v ∈ Vh
Ω
This was derived earlier under the assumption that u is sufficiently smooth. Since u is not very smooth at the corners of
Ω, one must justify carefully the integration by parts involving
u around the corners. This can be done by using the singular
function representation of u.
Since
Z
ah (uh , v) =
fv dx
∀ v ∈ Vh
Ω
by the definition of the C0 interior penalty method,
ah (u − uh , v) = 0
∀ v ∈ Vh
Two Mesh-Dependent Norms
kvk2h =
X
|v|2H 2 (T) +
T∈Th
|||v|||2h =
X
T∈Th
hh ii
X σ
∂v 2
|e| ∂n L2 (e)
e∈Eh
|v|2H 2 (T) +
hh ii
nn
oo
X σ
X |e| ∂v 2
∂ 2 v 2
+
|e| ∂n L2 (e)
σ
∂n2
L2 (e)
e∈Eh
e∈Eh
Two Mesh-Dependent Norms
kvk2h =
X
|v|2H 2 (T) +
T∈Th
|||v|||2h =
X
T∈Th
hh ii
X σ
∂v 2
|e| ∂n L2 (e)
e∈Eh
|v|2H 2 (T) +
hh ii
nn
oo
X σ
X |e| ∂v 2
∂ 2 v 2
+
|e| ∂n L2 (e)
σ
∂n2
L2 (e)
e∈Eh
e∈Eh
The first mesh-dependent norm k · kh is well-defined on H 2 (Ω).
The second mesh-dependent norm ||| · |||h is only well-defined if
v is a piecewise H s function for some s > 52 .
Two Mesh-Dependent Norms
kvk2h =
X
|v|2H 2 (T) +
T∈Th
|||v|||2h =
X
T∈Th
hh ii
X σ
∂v 2
|e| ∂n L2 (e)
e∈Eh
|v|2H 2 (T) +
hh ii
nn
oo
X σ
X |e| ∂v 2
∂ 2 v 2
+
|e| ∂n L2 (e)
σ
∂n2
L2 (e)
e∈Eh
e∈Eh
The first mesh-dependent norm k · kh is well-defined on H 2 (Ω).
The second mesh-dependent norm ||| · |||h is only well-defined if
v is a piecewise H s function for some s > 52 .
Since the solution u belongs to H 2+α (Ω) for α > 12 , the second
mesh dependent norm is well-defined on the space
hui + Vh
Two Mesh-Dependent Norms
kvk2h =
X
|v|2H 2 (T) +
T∈Th
|||v|||2h =
X
T∈Th
hh ii
X σ
∂v 2
|e| ∂n L2 (e)
e∈Eh
|v|2H 2 (T) +
hh ii
nn
oo
X σ
X |e| ∂v 2
∂ 2 v 2
+
|e| ∂n L2 (e)
σ
∂n2
L2 (e)
e∈Eh
e∈Eh
Relations between the Two Norms
Obviously
kvkh ≤ |||v|||h
∀ v ∈ Vh
Two Mesh-Dependent Norms
kvk2h =
X
|v|2H 2 (T) +
T∈Th
|||v|||2h =
X
hh ii
X σ
∂v 2
|e| ∂n L2 (e)
e∈Eh
|v|2H 2 (T) +
T∈Th
hh ii
nn
oo
X σ
X |e| ∂v 2
∂ 2 v 2
+
|e| ∂n L2 (e)
σ
∂n2
L2 (e)
e∈Eh
e∈Eh
Relations between the Two Norms
Obviously
kvkh ≤ |||v|||h
∀ v ∈ Vh
On the other hand
|||v|||h ≤ Ckvkh
because
∀ v ∈ Vh
X |e|
X
k{{∂ 2 v/∂n2 }}k2L2 (e) ≤ C
|v|2H 2 (T)
σ
e∈Eh
T∈Th
Two Mesh-Dependent Norms
kvk2h =
X
|v|2H 2 (T) +
T∈Th
|||v|||2h =
X
hh ii
X σ
∂v 2
|e| ∂n L2 (e)
e∈Eh
|v|2H 2 (T) +
T∈Th
hh ii
nn
oo
X σ
X |e| ∂v 2
∂ 2 v 2
+
|e| ∂n L2 (e)
σ
∂n2
L2 (e)
e∈Eh
e∈Eh
Relations between the Two Norms
Obviously
kvkh ≤ |||v|||h
∀ v ∈ Vh
On the other hand
|||v|||h ≤ Ckvkh
∀ v ∈ Vh
Therefore the two norms are equivalent on Vh .
Properties of ah (·, ·)
Properties of ah (·, ·)
Boundedness
ah (w, v) =
XZ
T∈Th
D2 w : D2 v dx +
T
n ∂ 2 w oohh ∂v ii
XZ n
ds
∂n2
∂n
e
e∈Eh
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
Properties of ah (·, ·)
Boundedness
ah (w, v) =
XZ
T∈Th
D2 w : D2 v dx +
T
n ∂ 2 w oohh ∂v ii
XZ n
ds
∂n2
∂n
e
e∈Eh
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
XZ
T∈Th
T
D2 w : D2 v dx ≤
X
T∈Th
|w|2H 2 (T)
1 X
2
T∈Th
|v|2H 2 (T)
1
2
Properties of ah (·, ·)
Boundedness
ah (w, v) =
XZ
T∈Th
D2 w : D2 v dx +
T
n ∂ 2 w oohh ∂v ii
XZ n
ds
∂n2
∂n
e
e∈Eh
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
|e| e ∂n
∂n
e∈Eh
≤
X σ hh ∂w ii2 1 X σ hh ∂v ii2 1
2
2
|e| ∂n L2 (e)
|e| ∂n L2 (e)
e∈Eh
e∈Eh
Properties of ah (·, ·)
Boundedness
ah (w, v) =
XZ
T∈Th
D2 w : D2 v dx +
T
n ∂ 2 w oohh ∂v ii
XZ n
ds
∂n2
∂n
e
e∈Eh
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
X Z nn ∂ 2 w oohh ∂v ii
ds
∂n2
∂n
e
e∈Eh
≤
X |e| n
n
o
o
1 X σ hh ∂v ii2 1
2
2
∂ 2 w 2
σ
∂n2
|e| ∂n L2 (e)
L2 (e)
e∈Eh
e∈Eh
Properties of ah (·, ·)
Boundedness
ah (w, v) =
XZ
T∈Th
D2 w : D2 v dx +
T
n ∂ 2 w oohh ∂v ii
XZ n
ds
∂n2
∂n
e
e∈Eh
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
X Z nn ∂ 2 v oohh ∂w ii
ds
∂n2
∂n
e
e∈Eh
≤
X |e| n
n
o
o
1 X σ hh ∂w ii2 1
2
2
∂ 2 v 2
σ
∂n2
|e| ∂n L2 (e)
L2 (e)
e∈Eh
e∈Eh
Properties of ah (·, ·)
Boundedness
ah (w, v) =
XZ
T∈Th
D2 w : D2 v dx +
T
n ∂ 2 w oohh ∂v ii
XZ n
ds
∂n2
∂n
e
e∈Eh
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
ah (w, v) ≤ 2|||w|||h |||v|||h
∀ v, w ∈ hui + Vh
Properties of ah (·, ·)
Boundedness
ah (w, v) =
XZ
T∈Th
D2 w : D2 v dx +
T
n ∂ 2 w oohh ∂v ii
XZ n
ds
∂n2
∂n
e
e∈Eh
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
ah (w, v) ≤ 2|||w|||h |||v|||h
∀ v, w ∈ hui + Vh
In particular
(∗)
ah (w, v) ≤ Ckwkh kvkh
∀ v, w ∈ Vh
Properties of ah (·, ·)
Boundedness
ah (w, v) =
XZ
T∈Th
D2 w : D2 v dx +
T
n ∂ 2 w oohh ∂v ii
XZ n
ds
∂n2
∂n
e
e∈Eh
n ∂2v o
ohh ∂w ii
XZ n
+
ds
∂n2
∂n
e
e∈Eh
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
ah (w, v) ≤ 2|||w|||h |||v|||h
∀ v, w ∈ hui + Vh
In particular
(∗)
ah (w, v) ≤ Ckwkh kvkh
∀ v, w ∈ Vh
But the estimate (∗) is not valid for v, w ∈ hui + Vh .
Properties of ah (·, ·)
Coercivity
We know that
1
ah (v, v) ≥ kvk2h
2
provided σ is sufficiently large.
∀ v ∈ Vh
Properties of ah (·, ·)
Coercivity
We know that
1
ah (v, v) ≥ kvk2h
2
∀ v ∈ Vh
provided σ is sufficiently large.
Therefore ah (·, ·) is also coercive with respect to the norm |||·|||h ,
i.e.,
ah (v, v) ≥ C|||v|||2h
∀ v ∈ Vh
Properties of ah (·, ·)
Coercivity
We know that
1
ah (v, v) ≥ kvk2h
2
∀ v ∈ Vh
provided σ is sufficiently large.
Therefore ah (·, ·) is also coercive with respect to the norm |||·|||h ,
i.e.,
ah (v, v) ≥ C|||v|||2h
∀ v ∈ Vh
Consequently
kvkh ≈ |||v|||h ≈
p
ah (v, v)
∀ v ∈ Vh
Error Estimate in the Energy Norm ||| · |||h
Error Estimate in the Energy Norm ||| · |||h
For any v ∈ Vh
|||u − uh |||h ≤ |||u − v|||h + |||v − uh |||h
Error Estimate in the Energy Norm ||| · |||h
For any v ∈ Vh
|||u − uh |||h ≤ |||u − v|||h + |||v − uh |||h
ah (v − uh , w)
≤ |||u − v|||h + C max
w∈Vh
|||w|||h
Coercivity
|||v|||2h ≤ Cah (v, v)
ah (v, v)
|||v|||h
ah (v, w)
=⇒|||v|||h ≤ C max
w∈Vh |||w|||h
=⇒|||v|||h ≤ C
Error Estimate in the Energy Norm ||| · |||h
For any v ∈ Vh
|||u − uh |||h ≤ |||u − v|||h + |||v − uh |||h
ah (v − uh , w)
≤ |||u − v|||h + C max
w∈Vh
|||w|||h
ah (v − u, w)
= |||u − v|||h + C max
w∈Vh
|||w|||h
Galerkin Orthogonality
ah (u − uh , w) = 0
∀ w ∈ Vh
Error Estimate in the Energy Norm ||| · |||h
For any v ∈ Vh
|||u − uh |||h ≤ |||u − v|||h + |||v − uh |||h
ah (v − uh , w)
≤ |||u − v|||h + C max
w∈Vh
|||w|||h
ah (v − u, w)
= |||u − v|||h + C max
w∈Vh
|||w|||h
≤ |||u − v|||h + C|||v − u|||h
Boundedness
ah (v − u, w) ≤ 2|||v − u|||h |||w|||h
(We cannot use the norm k · kh in this step.)
Error Estimate in the Energy Norm ||| · |||h
For any v ∈ Vh
|||u − uh |||h ≤ |||u − v|||h + |||v − uh |||h
ah (v − uh , w)
≤ |||u − v|||h + C max
w∈Vh
|||w|||h
ah (v − u, w)
= |||u − v|||h + C max
w∈Vh
|||w|||h
≤ |||u − v|||h + C|||v − u|||h
≤ C|||u − v|||h
Error Estimate in the Energy Norm ||| · |||h
For any v ∈ Vh
|||u − uh |||h ≤ |||u − v|||h + |||v − uh |||h
ah (v − uh , w)
≤ |||u − v|||h + C max
w∈Vh
|||w|||h
ah (v − u, w)
= |||u − v|||h + C max
w∈Vh
|||w|||h
≤ |||u − v|||h + C|||v − u|||h
≤ C|||u − v|||h
Therefore
|||u − uh |||h ≤ C inf |||u − v|||h
v∈Vh
Error Estimate in the Energy Norm ||| · |||h
Let Πh be the nodal interpolation operator from C(Ω̄) into Vh .
Standard Interpolation Error Estimates
−2(2+α)
hT
−2(1+α)
kζ − Πh ζk2L2 (T) + hT
|ζ − Πh ζ|2H 1 (T)
+ h−2α
|ζ − Πh ζ|2H 2 (T) ≤ Ckζk2H 2+α (T)
T
∀ T ∈ Th , ζ ∈ H 2+α (T)
Error Estimate in the Energy Norm ||| · |||h
Let Πh be the nodal interpolation operator from C(Ω̄) into Vh .
Standard Interpolation Error Estimates
−2(2+α)
hT
−2(1+α)
kζ − Πh ζk2L2 (T) + hT
|ζ − Πh ζ|2H 1 (T)
+ h−2α
|ζ − Πh ζ|2H 2 (T) ≤ Ckζk2H 2+α (T)
T
|||u − Πh u|||2h ≤
X
|u − Πh u|2H 2 (T) +
T∈Th
∀ T ∈ Th , ζ ∈ H 2+α (T)
hh
ii
X σ
∂(u − Πh u) 2
|e|
∂n
L2 (e)
e∈Eh
n
n
o
o
X |e| ∂ 2 (u − Πh u) 2
+
≤ Ch2α |u|2H 2+α (Ω)
σ
∂n2
L2 (e)
e∈Eh
Error Estimate in the Energy Norm ||| · |||h
Let Πh be the nodal interpolation operator from C(Ω̄) into Vh .
Standard Interpolation Error Estimates
−2(2+α)
hT
−2(1+α)
kζ − Πh ζk2L2 (T) + hT
|ζ − Πh ζ|2H 1 (T)
+ h−2α
|ζ − Πh ζ|2H 2 (T) ≤ Ckζk2H 2+α (T)
T
|||u − Πh u|||2h ≤
X
|u − Πh u|2H 2 (T) +
T∈Th
∀ T ∈ Th , ζ ∈ H 2+α (T)
hh
ii
X σ
∂(u − Πh u) 2
|e|
∂n
L2 (e)
e∈Eh
n
n
o
o
X |e| ∂ 2 (u − Πh u) 2
+
≤ Ch2α |u|2H 2+α (Ω)
σ
∂n2
L2 (e)
e∈Eh
Hence
|||u − uh |||h ≤ C inf |||u − v|||h
v∈Vh
≤ C|||u − Πh u|||h ≤ Chα |u|H 2+α (Ω) ≤ Chα kf kL2 (Ω)
Other Boundary Conditions
In the standard approach we need to use the mesh-dependent
norm ||| · |||h defined by
|||v|||2h =
X
T∈Th
|v|2H 2 (T) +
hh ii
nn
oo
X |e| X σ
∂v 2
∂ 2 v 2
+
|e| ∂n L2 (e)
σ
∂n2
L2 (e)
e∈Eh
e∈Eh
in order to exploit the Galerkin orthogonality.
But for the biharmonic equation with boundary conditions of
simply supported plates or the Cahn-Hilliard type, the solution
u may belong to H 2+α (Ω) for α ∈ (0, 12 ), in which case the norm
||| · |||h is not well-defined on hui + Vh .
Therefore the standard approach is problematic for these problems.
Error Estimate for the Post-Processed Solution
|u − Eh uh |H 2 (Ω) ≤ Chα kf kL2 (Ω)
Error Estimate for the Post-Processed Solution
|u − Eh uh |H 2 (Ω) ≤ Chα kf kL2 (Ω)
|u − Eh uh |H 2 (Ω) ≤ |u − Eh Πh u|H 2 (Ω) + |Eh (Πh u − uh )|H 2 (Ω)
≤ C hα |u|H 2+α (Ω) + |Πh u − uh |H 2 (Ω;Th )
≤ C hα |u|H 2+α (Ω) + |Πh u − u|H 2 (Ω;Th )
+ |u − uh |H 2 (Ω;Th )
≤ C hα |u|H 2+α (Ω) + |||u − uh |||h
≤ Chα |u|H 2+α (Ω)
≤ Chα kf kL2 (Ω)
Error Estimates in Lower Order Norms
Error Estimates in Lower Order Norms
For simplicity we assume that Ω is a convex polygon so that we
can take the index of elliptic regularity α to be 1, i.e., u ∈ H 3 (Ω)
and
|||u − uh |||h ≤ Chkf kL2 (Ω)
Error Estimates in Lower Order Norms
For simplicity we assume that Ω is a convex polygon so that we
can take the index of elliptic regularity α to be 1, i.e., u ∈ H 3 (Ω)
and
|||u − uh |||h ≤ Chkf kL2 (Ω)
Let F ∈ H −1 (Ω) = [H01 (Ω)]0 and ζ ∈ H02 (Ω) satisfy the clamped
plate problem
a(ζ, v) = F(v)
∀ v ∈ H02 (Ω)
Z
a(w, v) =
D2 w : D2 v dx
Ω
Error Estimates in Lower Order Norms
For simplicity we assume that Ω is a convex polygon so that we
can take the index of elliptic regularity α to be 1, i.e., u ∈ H 3 (Ω)
and
|||u − uh |||h ≤ Chkf kL2 (Ω)
Let F ∈ H −1 (Ω) = [H01 (Ω)]0 and ζ ∈ H02 (Ω) satisfy the clamped
plate problem
a(ζ, v) = F(v)
∀ v ∈ H02 (Ω)
Then we have the Shift Theorem
|ζ|H 3 (Ω) ≤ CΩ kFkH −1 (Ω)
Error Estimates in Lower Order Norms
For simplicity we assume that Ω is a convex polygon so that we
can take the index of elliptic regularity α to be 1, i.e., u ∈ H 3 (Ω)
and
|||u − uh |||h ≤ Chkf kL2 (Ω)
Let F ∈ H −1 (Ω) = [H01 (Ω)]0 and ζ ∈ H02 (Ω) satisfy the clamped
plate problem
a(ζ, v) = F(v)
∀ v ∈ H02 (Ω)
Then we have the Shift Theorem
|ζ|H 3 (Ω) ≤ CΩ kFkH −1 (Ω)
We can use this Shift Theorem to derive an error estimate for
|u − uh |H 1 (Ω) through a duality argument.
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
Let ζ ∈ H02 (Ω) satisfy
Z
D2 ζ : D2 v dx = F(v)
Ω
F(u − uh )
kFkH −1 (Ω)
∀ v ∈ H02 (Ω)
and ζh ∈ Vh satisfy
ah (ζh , v) = F(v)
∀ v ∈ Vh
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
Let ζ ∈ H02 (Ω) satisfy
Z
D2 ζ : D2 v dx = F(v)
∀ v ∈ H02 (Ω)
Ω
and ζh ∈ Vh satisfy
ah (ζh , v) = F(v)
∀ v ∈ Vh
Then ζ ∈ H 3 (Ω) by the Shift Theorem,
|ζ|H 3 (Ω) ≤ CkFkH −1 (Ω)
and
ah (ζ − ζh , v) = 0
∀ v ∈ Vh
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh ) = F(u) − F(uh )
F(u − uh )
kFkH −1 (Ω)
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
F(u − uh ) = F(u) − F(uh )
= a(ζ, u) − ah (ζh , uh )
a(ζ, v) = F(v)
ah (ζh , v) = F(v)
∀ v ∈ H02 (Ω)
∀ v ∈ Vh
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
F(u − uh ) = F(u) − F(uh )
= a(ζ, u) − ah (ζh , uh )
= ah (ζ, u) − ah (ζ, uh )
ah (v, w) = a(v, w)
ah (ζ − ζh , v) = 0
∀ v, w ∈ H02 (Ω) ∩ H 3 (Ω)
∀ v ∈ Vh
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
F(u − uh ) = F(u) − F(uh )
= a(ζ, u) − ah (ζh , uh )
= ah (ζ, u) − ah (ζ, uh )
= ah (ζ, u − uh )
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
F(u − uh ) = F(u) − F(uh )
= a(ζ, u) − ah (ζh , uh )
= ah (ζ, u) − ah (ζ, uh )
= ah (ζ, u − uh )
= ah (ζ − Πh ζ, u − uh )
ah (u − uh , v) = 0
∀ v ∈ Vh
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
F(u − uh ) = F(u) − F(uh )
= a(ζ, u) − ah (ζh , uh )
= ah (ζ, u) − ah (ζ, uh )
= ah (ζ, u − uh )
= ah (ζ − Πh ζ, u − uh )
≤ C|||ζ − Πh ζ|||h |||u − uh |||h
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
F(u − uh ) = F(u) − F(uh )
= a(ζ, u) − ah (ζh , uh )
= ah (ζ, u) − ah (ζ, uh )
= ah (ζ, u − uh )
= ah (ζ − Πh ζ, u − uh )
≤ C|||ζ − Πh ζ|||h |||u − uh |||h
≤ Ch|ζ|H 3 (Ω) hkf kL2 (Ω)
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
F(u − uh ) = F(u) − F(uh )
= a(ζ, u) − ah (ζh , uh )
= ah (ζ, u) − ah (ζ, uh )
= ah (ζ, u − uh )
= ah (ζ − Πh ζ, u − uh )
≤ C|||ζ − Πh ζ|||h |||u − uh |||h
≤ Ch|ζ|H 3 (Ω) hkf kL2 (Ω)
≤ Ch2 kFkH −1 (Ω) kf kL2 (Ω)
Error Estimates in Lower Order Norms
|u − uh |H 1 (Ω) =
sup
F∈H −1 (Ω)
F(u − uh )
kFkH −1 (Ω)
F(u − uh ) = F(u) − F(uh )
= a(ζ, u) − ah (ζh , uh )
= ah (ζ, u) − ah (ζ, uh )
= ah (ζ, u − uh )
= ah (ζ − Πh ζ, u − uh )
≤ C|||ζ − Πh ζ|||h |||u − uh |||h
≤ Ch|ζ|H 3 (Ω) hkf kL2 (Ω)
≤ Ch2 kFkH −1 (Ω) kf kL2 (Ω)
Therefore
|u − uh |H 1 (Ω) ≤ Ch2 kf kL2 (Ω)
Error Estimates in Lower Order Norms
We can also derive an error estimate for the post-processed
solution Eh uh in the H 1 norm.
Error Estimates in Lower Order Norms
We can also derive an error estimate for the post-processed
solution Eh uh in the H 1 norm.
Lemma
|Eh v|H 1 (Ω) ≈ |v|H 1 (Ω)
∀ v ∈ Vh
Error Estimates in Lower Order Norms
We can also derive an error estimate for the post-processed
solution Eh uh in the H 1 norm.
Lemma
|Eh v|H 1 (Ω) ≈ |v|H 1 (Ω)
∀ v ∈ Vh
|u − Eh u|H 1 (Ω) ≤ |u − Eh Πh u|H 1 (Ω) + |Eh (Πh u − uh )|H 1 (Ω)
≤ C h2 |u|H 3 (Ω) + |Πh u − uh |H 1 (Ω)
≤ C h2 |u|H 3 (Ω) + |Πh u − u|H 1 (Ω) + |u − uh |H 1 (Ω)
≤ C h2 |u|H 3 (Ω) + h2 kf kL2 (Ω)
≤ Ch2 kf kL2 (Ω)
Error Estimates in Lower Order Norms
We can also derive an error estimate for the post-processed
solution Eh uh in the H 1 norm.
Lemma
|Eh v|H 1 (Ω) ≈ |v|H 1 (Ω)
∀ v ∈ Vh
Proof.
|Eh v|H 1 (Ω) ≤ |Eh v − v|H 1 (Ω) + |v|H 1 (Ω)
≤ Ch|v|H 2 (Ω;Th ) + |v|H 1 (Ω)
≤ C|v|H 1 (Ω)
Error Estimates in Lower Order Norms
We can also derive an error estimate for the post-processed
solution Eh uh in the H 1 norm.
Lemma
|Eh v|H 1 (Ω) ≈ |v|H 1 (Ω)
∀ v ∈ Vh
Proof.
|Eh v|H 1 (Ω) ≤ |Eh v − v|H 1 (Ω) + |v|H 1 (Ω)
≤ Ch|v|H 2 (Ω;Th ) + |v|H 1 (Ω)
≤ C|v|H 1 (Ω)
|v|H 1 (Ω) = |Πh Eh v|H 1 (Ω)
= |Πh Eh v − Eh v|H 1 (Ω) + |Eh v|H 1 (Ω)
≤ Ch|Eh v|H 2 (Ω) + |Eh v|H 1 (Ω)
≤ C|Eh v|H 1 (Ω)
Error Estimates in Lower Order Norms
For a general polygonal domain with elliptic regularity index α ∈
( 12 , 2], we have
|u − uh |H 2−α (Ω) ≤ Ch2α kf kL2 (Ω)
|u − Eh u|H 2−α (Ω) ≤ Ch2α kf kL2 (Ω)
Error Estimates in Lower Order Norms
For a general polygonal domain with elliptic regularity index α ∈
( 12 , 2], we have
|u − uh |H 2−α (Ω) ≤ Ch2α kf kL2 (Ω)
|u − Eh u|H 2−α (Ω) ≤ Ch2α kf kL2 (Ω)
It follows from the Sobolev Embedding Theorem that
ku − uh kL∞ (Ω) ≤ Ch2α kf kL2 (Ω)
ku − Eh ukL∞ (Ω) ≤ Ch2α kf kL2 (Ω)
Reference
• Brenner and S.
C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains
J. Sci. Comput., 2005
Medius Error Analysis
Convergence of C1 Finite Element Methods
If we solve the clamped plate problem using a C1 finite element
space Vh ⊂ H02 (Ω), then it follows from the Galerkin orthogonality
a(u − uh , v) = 0
∀ v ∈ Vh
that, for any v ∈ Vh ,
|u − uh |2H 2 (Ω) = a(u − uh , u − uh )
= a(u − uh , u − v) + a(u − uh , v − uh )
= a(u − uh , u − v)
≤ |u − uh |H 2 (Ω) |u − v|H 2 (Ω)
and hence
|u − uh |H 1 (Ω) ≤ |u − v|H 1 (Ω)
∀ v ∈ Vh
Convergence of C1 Finite Element Methods
|u − uh |H 2 (Ω) ≤ inf |u − v|H 2 (Ω)
v∈Vh
Convergence of C1 Finite Element Methods
|u − uh |H 2 (Ω) ≤ inf |u − v|H 2 (Ω)
v∈Vh
The derivation of this estimate only requires that u ∈ H02 (Ω)
satisfies the variational/weak problem for the clamped plate.
Convergence of C1 Finite Element Methods
|u − uh |H 2 (Ω) ≤ inf |u − v|H 2 (Ω)
v∈Vh
The derivation of this estimate only requires that u ∈ H02 (Ω)
satisfies the variational/weak problem for the clamped plate.
Since smooth solutions are dense in H02 (Ω), this estimate implies that conforming finite element methods always converge,
without using any additional knowledge on the elliptic regularity
of u.
Convergence of C1 Finite Element Methods
|u − uh |H 2 (Ω) ≤ inf |u − v|H 2 (Ω)
v∈Vh
The derivation of this estimate only requires that u ∈ H02 (Ω)
satisfies the variational/weak problem for the clamped plate.
Since smooth solutions are dense in H02 (Ω), this estimate implies that conforming finite element methods always converge,
without using any additional knowledge on the elliptic regularity
of u.
On the other hand, to show that the C0 interior penalty method
converges by the standard approach, we must know some additional elliptic regularity of u. Otherwise we cannot use the
mesh-dependent norm ||| · |||h on u.
Convergence of C1 Finite Element Methods
|u − uh |H 2 (Ω) ≤ inf |u − v|H 2 (Ω)
v∈Vh
The derivation of this estimate only requires that u ∈ H02 (Ω)
satisfies the variational/weak problem for the clamped plate.
Since smooth solutions are dense in H02 (Ω), this estimate implies that conforming finite element methods always converge,
without using any additional knowledge on the elliptic regularity
of u.
On the other hand, to show that the C0 interior penalty method
converges by the standard approach, we must know some additional elliptic regularity of u. Otherwise we cannot use the
mesh-dependent norm ||| · |||h on u.
Question
Can we prove the convergence of C0 interior
penalty methods without using any additional elliptic regularity
of u?
Goal
Let uh be the solution of a C0 interior penalty method for the
biharmonic equation with the boundary conditions of clamped
plates. We want to derive an estimate for u − uh using only the
fact that u ∈ H02 (Ω) satisfies the variational/weak formulation of
the boundary value problem.
Goal
Let uh be the solution of a C0 interior penalty method for the
biharmonic equation with the boundary conditions of clamped
plates. We want to derive an estimate for u − uh using only the
fact that u ∈ H02 (Ω) satisfies the variational/weak formulation of
the boundary value problem.
I
We will use the mesh-dependent norm
kvkh =
X
|v|2H 2 (T) +
T∈Th
that is well-defined on H 2 (Ω).
hh ii 1
X σ
2
∂v 2
|e| ∂n L( e)
e∈Eh
Goal
Let uh be the solution of a C0 interior penalty method for the
biharmonic equation with the boundary conditions of clamped
plates. We want to derive an estimate for u − uh using only the
fact that u ∈ H02 (Ω) satisfies the variational/weak formulation of
the boundary value problem.
I
We will use the mesh-dependent norm
kvkh =
X
|v|2H 2 (T) +
T∈Th
hh ii 1
X σ
2
∂v 2
|e| ∂n L( e)
e∈Eh
that is well-defined on H 2 (Ω).
I
We will not perform integration by parts that involve u.
Goal
Main Theorem
ku − uh kh ≤ C inf ku − vkh + Osc(f )
v∈Vh
where
Osc(f ) =
X
T∈Th
is of higher order.
h4T
inf
q∈Pk−2 (T)
kf − qk2L2 (T)
1/2
(k = degree of the polynomials in Vh )
(quasi-optimal up to a higher order term)
Goal
Main Theorem
ku − uh kh ≤ C inf ku − vkh + Osc(f )
v∈Vh
where
Osc(f ) =
X
T∈Th
is of higher order.
h4T
inf
q∈Pk−2 (T)
kf − qk2L2 (T)
1/2
(k = degree of the polynomials in Vh )
Corollary
lim ku − uh kh = 0
h↓0
An Integration by Parts Formula
Let v and w be finite element functions that vanish on ∂Ω.
XZ
D2 w : D2 v dx
T∈Th
T
=
XZ
T∈Th
+
(∆2 w)v dx −
T
T∈Th
XZ
T∈Th
XZ
∂∆w ∂T
∂
∇w · ∇v ds
∂T ∂n
∂n
v ds
An Integration by Parts Formula
Let v and w be finite element functions that vanish on ∂Ω.
XZ
D2 w : D2 v dx
T∈Th
T
=
XZ
T∈Th
(∆2 w)v dx −
T
XZ
T∈Th
∂T
∂∆w ∂n
XZ
v ds
∂
+
∇w · ∇v ds
∂n
T∈Th ∂T
Z
X
X Z ∂∆w 2
v ds
=
(∆ w)v dx −
∂n
T∈Th T
T∈Th ∂T
X Z ∂ 2 w ∂v X Z ∂ 2 w ∂v +
ds +
ds
2
∂n
∂t
∂T ∂n
∂T ∂n∂t
T∈Th
T∈Th
An Integration by Parts Formula
XZ
T∈Th
D2 w : D2 v dx
T
=
XZ
T∈Th
+
2
(∆ w)v dx −
T
T∈Th
XZ
T∈Th
XZ
∂T
∂ 2 w ∂v ∂n2
∂n
∂∆w ∂T
ds +
∂n
v ds
XZ
T∈Th
∂T
∂ 2 w ∂v ∂n∂t
∂t
ds
An Integration by Parts Formula
XZ
T∈Th
D2 w : D2 v dx
T
=
XZ
T∈Th
+
2
(∆ w)v dx −
T
=
XZ
T∈Th
T∈Th
∂ 2 w ∂v XZ
T∈Th
XZ
∂T
∂n2
∂n
∂∆w ∂n
∂T
ds +
v ds
XZ
T∈Th
∂T
∂ 2 w ∂v ∂n∂t
∂t
ds
X Z hh ∂∆w ii
v ds
(∆ w)v dx +
∂n
T
e
i
2
e∈Eh
ohh ∂v ii
X Z hh ∂ 2 w iinn ∂v oo
X Z nn ∂ 2 w o
−
ds
−
ds
2
∂n2
∂n
∂n
e
e ∂n
i
e∈Eh
e∈Eh
X Z hh ∂ 2 w ii ∂v
−
ds
e ∂n∂t ∂t
i
e∈Eh
An Alternative Expression for ah (·, ·)
n ∂2w o
ohh ∂v ii
XZ n
ah (w, v) =
D w : D v dx +
ds
∂n2
∂n
T∈Th T
e∈Eh e
ohh ∂w ii
n ∂2v o
XZ n
ds
+
∂n2
∂n
e∈Eh e
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
XZ
2
2
e∈Eh
v and w are finite element
functions that vanish on ∂Ω.
An Alternative Expression for ah (·, ·)
n ∂2w o
ohh ∂v ii
XZ n
ah (w, v) =
D w : D v dx +
ds
∂n2
∂n
T∈Th T
e∈Eh e
ohh ∂w ii
n ∂2v o
XZ n
ds
+
∂n2
∂n
e∈Eh e
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
XZ
2
2
e∈Eh
v and w are finite element
functions that vanish on ∂Ω.
An Alternative Expression for ah (·, ·)
n ∂2w o
ohh ∂v ii
XZ n
ah (w, v) =
D w : D v dx +
ds
∂n2
∂n
T∈Th T
e∈Eh e
ohh ∂w ii
n ∂2v o
XZ n
ds
+
∂n2
∂n
e∈Eh e
X σ Z hh ∂w iihh ∂v ii
ds
+
|e| e ∂n
∂n
e∈Eh
n ∂2v o
ohh ∂w ii
XZ
XZ n
=
(∆2 w)v dx +
ds
∂n2
∂n
T
e
T∈Th
e∈Eh
hh ∂ 2 w iin
n ∂v o
o hh ∂ 2 w ii ∂v X Z hh ∂∆w ii
+
v−
−
ds
∂n
∂n2
∂n
∂n∂t ∂t
e
e∈Ehi
X σ Z hh ∂w iihh ∂v ii
+
ds
|e| e ∂n
∂n
XZ
2
e∈Eh
2
Strategy for Deriving the Main Theorem
Strategy for Deriving the Main Theorem
For any v ∈ Vh
ku − uh kh ≤ ku − vkh + kv − uh kh
ah (v − uh , w)
≤ ku − vkh + C max
v∈Vh
kwkh
Strategy for Deriving the Main Theorem
For any v ∈ Vh
ku − uh kh ≤ ku − vkh + kv − uh kh
ah (v − uh , w)
≤ ku − vkh + C max
v∈Vh
kwkh
If we can show that
max
v∈Vh
ah (v − uh , w)
≤ C ku − vkh + Osc(f )
kwkh
then
ku − uh kh ≤ C ku − vkh + Osc(f )
∀ v ∈ Vh
and hence
ku − uh kh ≤ C inf ku − vkh + Osc(f )
v∈Vh
Preliminary Estimates
Z
ah (v − uh , w) = ah (v, Eh w) + ah (v, w − Eh w) −
f w dx
Ω
Z
ah (uh , w) =
f w dx
Ω
Eh is an enriching operator that maps Vh into H02 (Ω).
Preliminary Estimates
Z
ah (v − uh , w) = ah (v, Eh w) + ah (v, w − Eh w) −
f w dx
Ω
First Term on the Right-Hand Side
n ∂ 2 (E w) oohh ∂v ii
XZ n
XZ
h
2
2
D v : D (Eh w) dx +
ah (v, Eh w) =
ds
∂n2
∂n
T
e
T∈Th
e∈Eh
Z
XZ
2
2
=
D (v − u) : D (Eh w) dx +
f (Eh w) dx
T∈Th
Ω
T
n ∂ 2 (E w) o
ohh ∂v ii
XZ n
h
ds.
+
∂n2
∂n
e
e∈Eh
Z
2
2
Z
D u : D (Eh w) dx =
Ω
f (Eh w) dx
Ω
since Eh w ∈ H02 (Ω)
Preliminary Estimates
Second Term on the Right-Hand side
ah (v, w − Eh w)
n ∂ 2 (w − E w) oohh ∂v ii
XZ n
XZ
h
(∆2 v)(w − Eh w)dx +
=
ds
2
∂n
∂n
T
e
T∈Th
e∈Eh
n ∂(w − E w) oo
hh ∂ 2 v iin
X Z hh ∂∆v ii
h
(w − Eh w) −
ds
+
2
∂n
∂n
∂n
e
i
e∈Eh
−
X Z hh ∂ 2 v ii ∂(w − Eh w)
ds
∂t
e ∂n∂t
i
e∈Eh
X 1 Z hh ∂v iihh ∂(w − Eh w) ii
+σ
ds
|e| e ∂n
∂n
e∈Eh
alternative expression for ah (·, ·)
Preliminary Estimates
ah (v − uh , w)
n ∂2w o
ohh ∂v ii
XZ
XZ n
2
2
=
D (v − u) : D (Eh w) dx +
ds
∂n2
∂n
T∈Th T
e∈Eh e
X σ Z hh ∂v iihh ∂(w − Eh w) ii
ds
+
|e| e ∂n
∂n
e∈Eh
X Z hh ∂ 2 v ii ∂(w − Eh w)
−
ds
∂t
e ∂n∂t
i
e∈Eh
hh ∂ 2 v iin
n ∂(w − E w) oo
X Z hh ∂∆v ii
h
+
(w − Eh w) −
ds
2
∂n
∂n
∂n
e
e∈Ehi
XZ
−
(f − ∆2 v)(w − Eh w)dx.
T∈Th
T
Preliminary Estimates
XZ
D2 (v − u) : D2 (Eh w) dx
T∈Th
T
≤
X
=
X
|u − v|2H 2 (T)
1 X
2
T∈Th
|Eh w|2H 2 (T)
T∈Th
|u − v|2H 2 (T)
1
2
|Eh w|H 2 (Ω)
T∈Th
≤C
X
|u − v|2H 2 (T)
1
2
kwkh
T∈Th
|Eh w|H 2 (Ω) ≤ C|w|H 2 (Ω;Th ) ≤ Ckwkh
1
2
Preliminary Estimates
X Z nn ∂ 2 w oohh ∂v ii X Z n
n ∂2w o
ohh ∂(v − u) ii ds = ds
2
2
∂n
∂n
∂n
∂n
e
e
e∈Eh
e∈Eh
X 1 hh ∂(u − v) ii2 1 X nn ∂ 2 w oo2 1
2
2
≤
|e|
2
|e|
∂n
∂n
L2 (e)
L2 (e)
e∈Eh
e∈Eh
X 1 hh ∂(u − v) ii2 1 X
1
2
2
≤C
|w|2H 2 (T)
|e|
∂n
L2 (e)
e∈Eh
T∈Th
X σ hh ∂(u − v) ii2 1
2
≤C
kwkh
|e|
∂n
L2 (e)
e∈Eh
X
e∈Eh
nn ∂ 2 w o
o2
X
|e|
≤
C
|w|2H 2 (T) ≤ Ckwk2h
∂n2
L2 (e)
T∈Th
Preliminary Estimates
X σ Z hh ∂v iihh ∂(w − E w) ii h
ds
|e| e ∂n
∂n
e∈Eh
≤C
X σ hh ∂(u − v) ii2 1
2
|e|
∂n
L2 (e)
e∈Eh
×
X σ hh ∂(w − E w) ii2 1
2
h
|e|
∂n
L2 (e)
e∈Eh
X σ hh ∂(u − v) ii2 1
2
≤C
kwkh
|e|
∂n
L2 (e)
e∈Eh
kwk2h =
X
T∈Th
|w|2H 2 (T) +
hh ii
X σ
∂w 2
|e| ∂n L2 (e)
e∈Eh
Preliminary Estimates
X Z hh ∂ 2 v ii ∂(w − E w) h
ds
∂n∂t
∂t
e
i
e∈Eh
≤
X
e∈Eh
hh ∂ 2 v ii2 1 X 1 ∂(w − E w) 2 1
2
2
h
|e|
∂n∂t L2 (e)
|e|
∂t
L2 (e)
i
e∈Eh
X 1 hh ∂v ii2 1
2
≤C
kwkh
|e| ∂n L2 (e)
i
e∈Eh
X σ hh ∂(u − v) ii2 1
2
≤C
kwkh
|e|
∂n
L2 (e)
i
e∈Eh
X 1
X
∂(w − Eh w) 2
2
2
≤C
h−2
T |w − Eh w|H 1 (T) ≤ Ckwkh
|e|
∂t
L2 (e)
i
e∈Eh
T∈Th
Preliminary Estimates
X Z hh ∂∆v ii
(w − Eh w) ds
∂n
e
i
e∈Eh
≤
hh ∂∆v ii2 1 X 1
1
2
2
2
|e|3 kw
−
E
wk
h
L2 (e)
3
∂n
|e|
L2 (e)
i
i
X
e∈Eh
e∈Eh
hh
X
ii2 1
2
3 ∂∆v ≤C
|e| kwkh
∂n
L2 (e)
i
e∈Eh
X 1
X
2
2
2
kw
−
E
wk
≤
C
h−4
h
T kw − Eh wkL2 (T) ≤ Ckwkh
L
(e)
3
2
|e|
i
e∈Eh
T∈Th
Preliminary Estimates
X Z hh ∂ 2 v iinn ∂(w − E w) o
o h
ds
2
∂n
∂n
e
i
e∈Eh
≤
hh ∂ 2 v ii2 1 X 1 n
1
n
oo
2
2
∂(w − Eh w) 2
|e|
2
∂n
|e|
∂n
L2 (e)
L2 (e)
i
i
X
e∈Eh
≤C
e∈Eh
hh ∂ 2 v ii2 1
2
|e|
kwkh
2
∂n
L2 (e)
i
X
e∈Eh
nn
o
o
X 1
X
∂(w − Eh w) 2
2
2
≤C
h−2
T |w−Eh w|H 1 (T) ≤ Ckwkh
|e|
∂n
L2 (e)
i
e∈Eh
T∈Th
Preliminary Estimates
XZ
(f − ∆2 v)(w − Eh w)dx
T∈Th
T
≤
X
≤
X
h4T kf − ∆2 vk2L2 (T)
1 X
2
T∈Th
T∈Th
T∈Th
h4T kf − ∆2 vk2L2 (T)
1
2
kwkh
2
h−4
T kw − Eh wkL2 (T)
1
2
Preliminary Estimates
ah (v − uh , w) ≤ C
X
|u − v|2H 2 (T) +
T∈Th
+
hh
ii
X σ
∂(u − v) 2
|e|
∂n
L2 (e)
e∈Eh
hh ∂ 2 v ii2
hh
ii2
X
3 ∂∆v |e|
|e| +
∂n2 L2 (e)
∂n
L2 (e)
i
i
X
e∈Eh
+
X
T∈Th
e∈Eh
h4T kf − ∆2 vk2L2 (T)
1
2
kwkh
Preliminary Estimates
ah (v − uh , w) ≤ C
X
|u − v|2H 2 (T) +
T∈Th
hh
ii
X σ
∂(u − v) 2
|e|
∂n
L2 (e)
e∈Eh
hh ∂ 2 v ii2
hh
ii2
X
3 ∂∆v |e|
|e| +
∂n2 L2 (e)
∂n
L2 (e)
i
i
X
+
e∈Eh
+
e∈Eh
X
h4T kf − ∆2 vk2L2 (T)
1
2
kwkh
T∈Th
Hence
max
v∈Vh
ah (v − uh , w)
kwkh
hh
ii
hh
ii2
X
X ∂ 2 v 2
3 ∂∆v +
|e|
≤ C ku − vk2h +
|e|
∂n2 L2 (e)
∂n
L2 (e)
i
i
e∈Eh
e∈Eh
+
X
T∈Th
h4T kf − ∆2 vk2L2 (T)
1
2
Preliminary Estimates
According to our strategy, it only remains to show that
hh
hh ∂ 2 v ii2
ii2
X
X
3 ∂∆v |e|
+
+
h4T kf − ∆2 vk2L2 (T)
|e|
2
∂n
∂n
L2 (e)
L2 (e)
T∈Th
e∈Ehi
e∈Ehi
≤ C ku − vk2h + Osc(f )2
X
Preliminary Estimates
According to our strategy, it only remains to show that
hh
hh ∂ 2 v ii2
ii2
X
X
3 ∂∆v |e|
+
+
h4T kf − ∆2 vk2L2 (T)
|e|
2
∂n
∂n
L2 (e)
L2 (e)
T∈Th
e∈Ehi
e∈Ehi
≤ C ku − vk2h + Osc(f )2
X
These are called efficiency estimates in a posteriori error analysis when v = uh , the solution of the discrete problem.
Local Efficiency Estimates
Local Efficiency Estimates
Estimate for h2T kf − ∆2 vkL2 (T)
Osc(f )2 =
X
T∈Th
=
X
h4T
inf
q∈Pk−2 (T)
kf − qk2L2 (T)
h4T kf − f̄ k2L2 (T)
T∈Th
f̄ T = L2 projection of f on Pk−2 (T)
Local Efficiency Estimates
Estimate for h2T kf − ∆2 vkL2 (T)
Osc(f )2 =
X
T∈Th
=
X
h4T
inf
q∈Pk−2 (T)
kf − qk2L2 (T)
h4T kf − f̄ k2L2 (T)
T∈Th
f̄ T = L2 projection of f on Pk−2 (T)
Let ζ ∈ P6 (T) be the polynomial in H02 (T) that equals 1 at the
center of T.
−1
|ζ|H 2 (T) ≤ Ch−2
T kζkL2 (T) ≤ ChT
ζ is known as a bubble function.
(Verfürth 1994)
Local Efficiency Estimates
Equivalence of norms on finite dimensional spaces
Z
Z
C1 (f̄ − ∆2 v)2 ζ 2 dx ≤ kf̄ − ∆2 vk2L2 (T) ≤ C2 (f̄ − ∆2 v)2 ζ dx
T
T
Local Efficiency Estimates
Equivalence of norms on finite dimensional spaces
Z
Z
C1 (f̄ − ∆2 v)2 ζ 2 dx ≤ kf̄ − ∆2 vk2L2 (T) ≤ C2 (f̄ − ∆2 v)2 ζ dx
T
T
Let the function z be defined by

(f̄ − ∆2 v)ζ
z=
0
on T
on Ω \ T
Then z ∈ H02 (Ω) and hence
Z
Z
Z
Z
2
2
2
2
D u : D z dx =
D u : D z dx =
f z dx =
f z dx
T
Ω
Ω
T
Local Efficiency Estimates
kf̄ − ∆2 vk2L2 (T) ≤ C
Z
kf̄ − ∆
(f̄ − ∆2 v)2 ζ dx
T
2
vk2L2 (T)
Z
≤ C2
T
(f̄ − ∆2 v)2 ζ dx
Local Efficiency Estimates
Z
kf̄ − ∆2 vk2L2 (T) ≤ C (f̄ − ∆2 v)2 ζ dx
T
Z
2
= C (f̄ − ∆ v)z dx
T
z = (f̄ − ∆2 v)ζ
Local Efficiency Estimates
Z
kf̄ − ∆2 vk2L2 (T) ≤ C (f̄ − ∆2 v)2 ζ dx
T
Z
2
= C (f̄ − ∆ v)z dx
T
Z
Z
hZ
i
2
=C
f z dx − (∆ v)z dx + (f̄ − f )z dx
T
T
T
Local Efficiency Estimates
Z
kf̄ − ∆2 vk2L2 (T) ≤ C (f̄ − ∆2 v)2 ζ dx
T
Z
2
= C (f̄ − ∆ v)z dx
T
Z
Z
hZ
i
2
=C
f z dx − (∆ v)z dx + (f̄ − f )z dx
T
T
T
Z
Z
hZ
i
2
2
2
2
=C
D u : D z dx − D v : D z dx + (f̄ − f )z dx
T
T
Z
2
2
T
Z
D u : D z dx =
ZT
T
D2 v : D2 z dx =
f z dx
ZT
T
(∆2 v)z dx
Local Efficiency Estimates
Z
kf̄ − ∆2 vk2L2 (T) ≤ C (f̄ − ∆2 v)2 ζ dx
T
Z
2
= C (f̄ − ∆ v)z dx
T
Z
Z
hZ
i
2
=C
f z dx − (∆ v)z dx + (f̄ − f )z dx
T
T
T
Z
Z
hZ
i
2
2
2
2
=C
D u : D z dx − D v : D z dx + (f̄ − f )z dx
T
T
T
Z
hZ
i
=C
(D2 u − D2 v) : D2 z dx + (f̄ − f )z dx
T
T
Local Efficiency Estimates
Z
kf̄ − ∆2 vk2L2 (T) ≤ C (f̄ − ∆2 v)2 ζ dx
T
Z
2
= C (f̄ − ∆ v)z dx
T
Z
Z
hZ
i
2
=C
f z dx − (∆ v)z dx + (f̄ − f )z dx
T
T
T
Z
Z
hZ
i
2
2
2
2
=C
D u : D z dx − D v : D z dx + (f̄ − f )z dx
T
T
T
Z
hZ
i
=C
(D2 u − D2 v) : D2 z dx + (f̄ − f )z dx
T
T
≤ C |u − v|H 2 (T) |z|H 2 (T) + kf − f̄ kL2 (T) kzkL2 (T)
Local Efficiency Estimates
Z
kf̄ − ∆2 vk2L2 (T) ≤ C (f̄ − ∆2 v)2 ζ dx
T
Z
2
= C (f̄ − ∆ v)z dx
T
Z
Z
hZ
i
2
=C
f z dx − (∆ v)z dx + (f̄ − f )z dx
T
T
T
Z
Z
hZ
i
2
2
2
2
=C
D u : D z dx − D v : D z dx + (f̄ − f )z dx
T
T
T
Z
hZ
i
=C
(D2 u − D2 v) : D2 z dx + (f̄ − f )z dx
T
T
≤ C |u − v|H 2 (T) |z|H 2 (T) + kf − f̄ kL2 (T) kzkL2 (T)
≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kzkL2 (T)
|z|H 2 (Ω) ≤ Ch−2
T kzkL2 (T)
Local Efficiency Estimates
Z
kf̄ − ∆2 vk2L2 (T) ≤ C (f̄ − ∆2 v)2 ζ dx
T
Z
2
= C (f̄ − ∆ v)z dx
T
Z
Z
hZ
i
2
=C
f z dx − (∆ v)z dx + (f̄ − f )z dx
T
T
T
Z
Z
hZ
i
2
2
2
2
=C
D u : D z dx − D v : D z dx + (f̄ − f )z dx
T
T
T
Z
hZ
i
=C
(D2 u − D2 v) : D2 z dx + (f̄ − f )z dx
T
T
≤ C |u − v|H 2 (T) |z|H 2 (T) + kf − f̄ kL2 (T) kzkL2 (T)
≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kzkL2 (T)
2
≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kf̄ − ∆ vkL2 (T)
C1 kzk2L2 (T)
Z
= C1
T
(f̄ − ∆2 v)2 ζ 2 dx ≤ kf̄ − ∆2 vk2L2 (T)
Local Efficiency Estimates
Z
kf̄ − ∆2 vk2L2 (T) ≤ C (f̄ − ∆2 v)2 ζ dx
T
Z
2
= C (f̄ − ∆ v)z dx
T
Z
Z
hZ
i
2
=C
f z dx − (∆ v)z dx + (f̄ − f )z dx
T
T
T
Z
Z
hZ
i
2
2
2
2
=C
D u : D z dx − D v : D z dx + (f̄ − f )z dx
T
T
T
Z
hZ
i
=C
(D2 u − D2 v) : D2 z dx + (f̄ − f )z dx
T
T
≤ C |u − v|H 2 (T) |z|H 2 (T) + kf − f̄ kL2 (T) kzkL2 (T)
≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kzkL2 (T)
2
≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kf̄ − ∆ vkL2 (T)
Local Efficiency Estimates
2
kf̄ − ∆2 vk2L2 (T) ≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kf̄ − ∆ vkL2 (T)
Local Efficiency Estimates
2
kf̄ − ∆2 vk2L2 (T) ≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kf̄ − ∆ vkL2 (T)
which implies
kf̄ − ∆2 vkL2 (T) ≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T)
Local Efficiency Estimates
2
kf̄ − ∆2 vk2L2 (T) ≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kf̄ − ∆ vkL2 (T)
which implies
kf̄ − ∆2 vkL2 (T) ≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T)
It follows that
kf − ∆2 vkL2 (T) ≤ kf − f̄ kL2 (T) + kf̄ − ∆2 vkL2 (T)
≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T)
Local Efficiency Estimates
2
kf̄ − ∆2 vk2L2 (T) ≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T) kf̄ − ∆ vkL2 (T)
which implies
kf̄ − ∆2 vkL2 (T) ≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T)
It follows that
kf − ∆2 vkL2 (T) ≤ kf − f̄ kL2 (T) + kf̄ − ∆2 vkL2 (T)
≤ C h−2
T |u − v|H 2 (T) + kf − f̄ kL2 (T)
and hence
h2T kf − ∆2 vkL2 (T) ≤ C |u − v|H 2 (T) + h2T kf − f̄ kL2 (T)
{z
}
|
local oscillation
Local Efficiency Estimates
1
Estimate for |e| 2 k[[∂ 2 v/∂n2 ]]kL2 (e)
1
|e| 2 k[[∂ 2 v/∂n2 ]]kL2 (e) ≤ C
X
T∈Te
|u − v|H 2 (T) + h2T kf − f̄ kL2 (T)
|
{z
}
local oscillation
3
2
Estimate for |e| k[[∂(∆v)/∂n]]k2L2 (e)
3
|e| 2 k[[∂(∆v)/∂n]]kL2 (e)
hh
ii
X
− 12 ∂(u − v) 2
≤C
|u − v|H 2 (T) + hT kf − f̄ kL2 (T) + |e| ∂n
L2 (e)
|
{z
}
T∈T
e
local oscillation
Te is the set of the triangles that share e as a common edge.
Local Efficiency Estimates
1
Estimate for |e| 2 k[[∂ 2 v/∂n2 ]]kL2 (e)
1
|e| 2 k[[∂ 2 v/∂n2 ]]kL2 (e) ≤ C
X
T∈Te
|u − v|H 2 (T) + h2T kf − f̄ kL2 (T)
|
{z
}
local oscillation
3
2
Estimate for |e| k[[∂(∆v)/∂n]]k2L2 (e)
3
|e| 2 k[[∂(∆v)/∂n]]kL2 (e)
hh
ii
X
− 12 ∂(u − v) 2
≤C
|u − v|H 2 (T) + hT kf − f̄ kL2 (T) + |e| ∂n
L2 (e)
|
{z
}
T∈T
e
local oscillation
Estimate for h2T kf − ∆2 vkL2 (T)
h2T kf − ∆2 vkL2 (T) ≤ C |u − v|H 2 (T) + h2T kf − f̄ kL2 (T)
|
{z
}
local oscillation
Local Efficiency Estimates
Summing up the local efficiency estimates over all the triangles
and edges we have
hh ∂ 2 v ii2
hh
ii2
X
X
3 ∂∆v h4T kf − ∆2 vk2L2 (T)
|e|
+
|e|
+
2
∂n
∂n
L2 (e)
L2 (e)
T∈Th
e∈Ehi
e∈Ehi
2
2
≤ C ku − vkh + Osc(f )
X
which completes the proof of the Main Theorem.
Concrete Error Estimate
ku − uh kh ≤ C inf ku − vkh + Osc(f )
v∈Vh
≤ C ku − Πh ukh + Osc(f )
≤ Chα kf kL2 (Ω)
Concrete Error Estimate
ku − uh kh ≤ C inf ku − vkh + Osc(f )
v∈Vh
≤ C ku − Πh ukh + Osc(f )
≤ Chα kf kL2 (Ω)
Error Estimate in ||| · |||h
Concrete Error Estimate
ku − uh kh ≤ C inf ku − vkh + Osc(f )
v∈Vh
≤ C ku − Πh ukh + Osc(f )
≤ Chα kf kL2 (Ω)
Error Estimate in ||| · |||h
|||u − uh |||h ≤ |||u − Πh u|||h + |||Πh u − uh |||h
≤ C hα kf kL2 (Ω) + kΠh u − uh kh
≤ C hα kf kL2 (Ω) + kΠh u − ukh + ku − uh kh
≤ Chα kf kL2 (Ω)
References
• Gudi, A new error analysis for discontinuous finite element
methods for linear elliptic problems
Math. Comp., 2010
• Brenner and Neilan, A C0 interior penalty method for a
fourth order elliptic singular perturbation problem
SIAM J. Numer. Anal., 2011
• Brenner, Gu, Gudi and S., A C0 interior penalty method for
linear fourth order boundary value problems with boundary
conditions of the Cahn-Hilliard type
SIAM J. Numer. Anal., 2012
• Gudi, Gupta and Nataraj, Analysis of an interior penalty
method for fourth order problems on polygonal domains
J. Sci. Comput., 2013
A Posteriori Error Analysis
A Residual Based Error Estimator
A Residual Based Error Estimator
Let T ∈ Th be arbitrary. The residual error
ηT = h2T kf − ∆2 uh kL2 (T)
measures the extent to which uh fails to satisfy the biharmonic
equation.
A Residual Based Error Estimator
Let T ∈ Th be arbitrary. The residual error
ηT = h2T kf − ∆2 uh kL2 (T)
measures the extent to which uh fails to satisfy the biharmonic
equation.
Let e ∈ Eh be arbitrary. The residual
ηe,1 =
hh
ii
σ ∂uh 1
∂n L2 (e)
|e| 2
measures the extent to which uh fails to be in H02 (Ω).
A Residual Based Error Estimator
The residual
ηe,2
hh ∂ 2 u ii
h = |e| ∂n2
L2 (e)
1
2
measures the extent that uh fails to be in H 3 (Ω).
A Residual Based Error Estimator
The residual
ηe,2
hh ∂ 2 u ii
h = |e| ∂n2
L2 (e)
1
2
measures the extent that uh fails to be in H 3 (Ω).
The residual
hh
ii
3 ∂(∆uh ) ηe,3 = |e| 2 ∂ne
L2 (e)
measures the extent that uh fails to be in H 4 (Ω).
A Residual Based Error Estimator
The residual-based error estimator ηh is defined by
ηh =
hX
T∈Th
ηT2 +
X
e∈Eh
2
ηe,1
+
X
e∈Ehi
2
2
ηe,2
+ ηe,3
i 12
A Residual Based Error Estimator
The residual-based error estimator ηh is defined by
ηh =
hX
ηT2 +
T∈Th
Remark
X
e∈Eh
2
ηe,1
+
X
2
2
ηe,2
+ ηe,3
i 12
e∈Ehi
We can replace the definition of ηe,3 by
3
|e| 2 [[∂ 3 uh /∂n3e ]]kL2 (e)
A Residual Based Error Estimator
The residual-based error estimator ηh is defined by
ηh =
hX
ηT2 +
T∈Th
Remark
X
e∈Eh
2
ηe,1
+
X
2
2
ηe,2
+ ηe,3
i 12
e∈Ehi
We can replace the definition of ηe,3 by
3
|e| 2 [[∂ 3 uh /∂n3e ]]kL2 (e)
Remark
The error estimator ηe,3 is identically 0 for the
quadratic C0 interior penalty method.
Local Efficiency
Local Efficiency
Adaptive Loop
Solve → Estimate → Mark → Refine
In the Estimate step we compute ηT , ηe,1 , ηe,2 and ηe,3 over all
the triangles and edges. It is important that if one of the residuals is large at a triangle or an edge, then the true error is also
large there. Consequently we know that we should refine the
mesh at such locations.
Local Efficiency
Adaptive Loop
Solve → Estimate → Mark → Refine
In the Estimate step we compute ηT , ηe,1 , ηe,2 and ηe,3 over all
the triangles and edges. It is important that if one of the residuals is large at a triangle or an edge, then the true error is also
large there. Consequently we know that we should refine the
mesh at such locations.
This property is known as the local efficiency of the error estimator.
Local Efficiency
Local Efficiency of ηT
ηT = h2T kf − ∆2 uh kL2 (T) ≤ C |u − uh |H 2 (T) + h2T kf − f̄ kL2 (T)
Local Efficiency
Local Efficiency of ηT
ηT = h2T kf − ∆2 uh kL2 (T) ≤ C |u − uh |H 2 (T) + h2T kf − f̄ kL2 (T)
Local Efficiency of ηe,1
ηe,1
1
hh
hh
ii
ii
√ σ2 σ ∂uh ∂(u − uh ) = σ
= 1
1
∂n L2 (e)
∂n
L2 (e)
|e| 2
|e| 2
Local Efficiency
Local Efficiency of ηT
ηT = h2T kf − ∆2 uh kL2 (T) ≤ C |u − uh |H 2 (T) + h2T kf − f̄ kL2 (T)
Local Efficiency of ηe,1
ηe,1
1
hh
hh
ii
ii
√ σ2 σ ∂uh ∂(u − uh ) = σ
= 1
1
∂n L2 (e)
∂n
L2 (e)
|e| 2
|e| 2
Local Efficiency of ηe,2
hh 2 ii
X
1 ∂ uh ηe,2 = |e| 2 ≤C
|u − uh |H 2 (T) + h2T kf − f̄ kL2 (T)
2
∂n
L2 (e)
T∈Te
Local Efficiency
Local Efficiency of ηe,3
ηe,3
hh ∂(∆u ) ii
h
= |e| ∂n
L2 (e)
X
≤C
|u − uh |H 2 (T) + h2T kf − f̄ kL2 (T)
3
2
T∈Te
σ 12 hh ∂(u − u ) ii
h
+
1
∂n
L2 (e)
|e| 2
Local Efficiency
Local Efficiency of ηe,3
ηe,3
hh ∂(∆u ) ii
h
= |e| ∂n
L2 (e)
X
≤C
|u − uh |H 2 (T) + h2T kf − f̄ kL2 (T)
3
2
T∈Te
σ 12 hh ∂(u − u ) ii
h
+
1
∂n
L2 (e)
|e| 2
The local efficiency of ηT , ηe,2 and ηe,3 have already been established in the medius analysis.
The local efficiency of ηe,1 is trivial.
Reliability
Reliability
Adaptive Loop
Solve → Estimate → Mark → Refine
By refining the mesh at the locations where the residuals are
large, we hope to reduce the error estimator and hence the true
error. This requires the true error to be bounded by the error
estimator.
Reliability
Adaptive Loop
Solve → Estimate → Mark → Refine
By refining the mesh at the locations where the residuals are
large, we hope to reduce the error estimator and hence the true
error. This requires the true error to be bounded by the error
estimator.
The property that the true error is bounded by the error estimator is known as the reliability of the error estimator.
Reliability
We want to show that
ku − uh kh ≤ Cηh
Reliability
We want to show that
ku − uh kh ≤ Cηh
Recall
ku − uh k2h =
X
T∈Th
|u − uh |2H 2 (T) +
hh
ii
X σ
∂(u − uh ) 2
|e|
∂n
L2 (e)
e∈Eh
Reliability
We want to show that
ku − uh kh ≤ Cηh
Recall
ku − uh k2h =
X
|u − uh |2H 2 (T) +
T∈Th
hh
ii
X σ
∂(u − uh ) 2
|e|
∂n
L2 (e)
e∈Eh
Since
hh
ii
hh
ii
X σ
X σ
X
∂(u − uh ) 2
∂uh 2
2
=
≤
ηe,1
|e|
∂n
|e|
∂n L2 (e)
L2 (e)
e∈Eh
We only need to bound
e∈Eh
P
T∈Th
|u − uh |2H 2 (T) .
e∈Eh
Another Enriching Operator
We can use averaging to construct an enriching operator
Eh : Vh −→ Wh (⊂ H02 (Ω))
where Wh is the finite element space defined by the HsiehClough-Tocher macro element.
C1 piecewise cubic polynomials
(12 dof)
Another Enriching Operator
Estimates for Eh
X
2
−2
2
2
h−4
T kv − Eh vkL2 (T) + hT |v − Eh v|H 1 (T) + |v − Eh v|H 2 (T)
T∈Th
≤C
hh ii
X 1
∂v 2
|e| ∂n L2 (e)
i
e∈Eh
Another Enriching Operator
Estimates for Eh
X
2
−2
2
2
h−4
T kv − Eh vkL2 (T) + hT |v − Eh v|H 1 (T) + |v − Eh v|H 2 (T)
T∈Th
≤C
hh ii
X 1
∂v 2
|e| ∂n L2 (e)
i
e∈Eh
Remark We can also use this enriching operator in the postprocessing procedure.
Another Enriching Operator
Estimates for Eh
X
2
−2
2
2
h−4
T kv − Eh vkL2 (T) + hT |v − Eh v|H 1 (T) + |v − Eh v|H 2 (T)
T∈Th
≤C
hh ii
X 1
∂v 2
|e| ∂n L2 (e)
i
e∈Eh
Remark We can also use this enriching operator in the postprocessing procedure.
Remark Since we are not using relatives, the operator Eh is
no longer one-to-one and hence not appropriate for the analysis
of multigrid algorithms.
Estimate for
P
T∈Th
|u − uh |2H2 (T)
Estimate for
X
P
T∈Th
|u − uh |2H2 (T)
|u − uh |2H 2 (T) ≤ 2
T∈Th
X
|u − Eh uh |2H 2 (T) + |uh − Eh uh |2H 2 (T)
T∈Th
≤ 2|u − Eh uh |2H 2 (Ω) + C
X
2
ηe,1
,
e∈Eh
X
T∈Th
|v − Eh v|2H 2 (T) ≤ C
hh ii
X 1
∂v 2
|e| ∂n L2 (e)
i
e∈Eh
Estimate for
X
T∈Th
P
T∈Th
|u − uh |2H2 (T)
|u − uh |2H 2 (T) ≤ 2
X
|u − Eh uh |2H 2 (T) + |uh − Eh uh |2H 2 (T)
T∈Th
≤ 2|u − Eh uh |2H 2 (Ω) + C
X
e∈Eh
It only remains to estimate
|u − Eh uh |H 2 (Ω)
2
ηe,1
,
Estimate for |u − Eh uh |H2 (Ω)
Estimate for |u − Eh uh |H2 (Ω)
By duality
|u − Eh uh |H 2 (Ω) =
a(u − Eh uh , φ)
|φ|H 2 (Ω)
φ∈H 2 (Ω)
sup
0
Estimate for |u − Eh uh |H2 (Ω)
By duality
|u − Eh uh |H 2 (Ω) =
a(u − Eh uh , φ)
|φ|H 2 (Ω)
φ∈H 2 (Ω)
sup
0
Strategy
Show that
a(u − Eh uh , φ)
≤ Cηh
|φ|H 2 (Ω)
φ∈H 2 (Ω)
sup
0
Estimate for |u − Eh uh |H2 (Ω)
By duality
|u − Eh uh |H 2 (Ω) =
a(u − Eh uh , φ)
|φ|H 2 (Ω)
φ∈H 2 (Ω)
sup
0
Z
a(u − Eh uh , φ) =
D2 (u − Eh uh ) : D2 φ dx
Ω
XZ
XZ
2
=
D (uh − Eh uh ) : D2 φ dx −
D2 uh : D2 (φ − Πh φ) dx
T
T∈Th
Z
T∈Th
D2 u : D2 φ dx −
+
Ω
=
XZ
T∈Th
XZ
D2 uh : D2 (Πh φ)dx
T
T∈Th
2
2
D (uh − Eh uh ) : D φ dx −
T
+ ah (uh , Πh φ) −
XZ
T∈Th
XZ
T∈Th
T
T
2
2
D2 uh : D2 (φ − Πh φ) dx
T
Z
D uh : D (Πh φ)dx +
f (φ − Πh φ)dx
Ω
Estimate for |u − Eh uh |H2 (Ω)
Note that the integration by parts formula
XZ
D2 w : D2 v dx
T∈Th
T
=
X Z hh ∂∆w ii
v ds
(∆ w)v dx +
∂n
T
e
i
XZ
T∈Th
2
e∈Eh
n ∂2w o
ohh ∂v ii
XZ n
X Z hh ∂ 2 w iinn ∂v oo
−
ds −
ds
2
∂n2
∂n
∂n
e
e ∂n
i
e∈Eh
e∈Eh
X Z hh ∂ 2 w ii ∂v
ds
−
e ∂n∂t ∂t
i
e∈Eh
from medius analysis is also valid for w ∈ Vh and v ∈ H02 (Ω).
Estimate for |u − Eh uh |H2 (Ω)
XZ
T∈Th
=
D2 uh : D2 (φ − Πh φ) dx
T
X Z hh ∂(∆uh ) ii
(φ − Πh φ)ds
(∆ uh )(φ − Πh φ)dx +
∂n
T
e
i
XZ
T∈Th
2
e∈Eh
X Z nn ∂ 2 uh oohh ∂Πh φ ii
X Z hh ∂ 2 uh iinn ∂(φ − Πh φ) oo
+
ds
−
ds
∂n2
∂n
∂n2
∂n
e
e
i
e∈Eh
e∈Eh
X Z hh ∂ 2 uh ii ∂(φ − Πh φ)
−
ds
∂t
e ∂n∂t
i
e∈Eh
Estimate for |u − Eh uh |H2 (Ω)
ah (uh , Πh φ) −
XZ
T∈Th
D2 uh : D2 (Πh φ)dx
T
n ∂ 2 Π φ oohh ∂u ii
ohh ∂Π φ ii
XZ n
X Z nn ∂ 2 uh o
h
h
h
ds
+
ds
=
∂n2
∂n
∂n2
∂n
e
e
e∈Eh
e∈Eh
X σ Z hh ∂uh iihh ∂Πh φ ii
+
ds.
|e| e ∂n
∂n
e∈Eh
Estimate for |u − Eh uh |H2 (Ω)
a(u − Eh uh , φ)
XZ
XZ
D2 (uh − Eh uh ) : D2 φ dx +
(f − ∆2 uh )(φ − Πh φ)dx
=
T
T∈Th
T∈Th
T
X Z hh ∂(∆uh ) ii
−
(φ − Πh φ)ds
∂n
e
i
e∈Eh
n ∂(φ − Π φ) o
o
X Z hh ∂ 2 uh iin
h
+
ds
∂n2
∂n
e
i
e∈Eh
X Z hh ∂ 2 uh ii ∂(φ − Πh φ)
X Z hh ∂uh iinn ∂ 2 Πh φ oo
+
ds +
ds
∂t
∂n
∂n2
e ∂n∂t
e∈Eh e
e∈Ehi
X σ Z hh ∂uh iihh ∂Πh φ ii
+
ds
|e| e ∂n
∂n
e∈Eh
Estimate for |u − Eh uh |H2 (Ω)
a(u − Eh uh ) ≤ Cηh |φ|H 2 (Ω)
which implies
a(u − Eh uh , φ)
≤ Cηh
|φ|H 2 (Ω)
φ∈H 2 (Ω)
sup
0
and completes the proof of reliability.
Numerical Results
We have implemented an adaptive algorithm
I
Solve: compute uh
I
Estimate: compute ηh
I
Mark: mark the triangles to be refined
I
Refine
using the error estimator ηh and the bulk marking strategy of
Dörfler, and tested it on an L-shaped domain using an exact
solution with the correct singularity.
bulk marking strategy
♣ choose a number θ between 0 and 1
♣ mark enough triangles so that the residuals associated
with them and their edges add up to more than θηh
Numerical Results
1
10
Error and Estimator
Error
Estimator
0
10
3
10
4
10
Degrees of Freedom
5
10
Numerical Results
1
10
Error and Estimator
Error on Uniform Mesh
Error on Adaptive Mesh
0
10
3
10
4
10
Degrees of Freedom
5
10
Open Problem
Prove the convergence and optimality of the adaptive algorithm.
References
• Brenner, Gudi and S., An a posteriori error estimator for
a quadratic C0 interior penalty method for the biharmonic
problem
IMA J. Numer. Anal., 2010
References
• Brenner, Gudi and S., An a posteriori error estimator for
a quadratic C0 interior penalty method for the biharmonic
problem
IMA J. Numer. Anal., 2010
• Cascon, Kreuzer, Nochetto and Siebert, Quasi-optimal
convergence rate for an adaptive finite element method
SIAM J. Numer. Anal., 2008
• Nochetto, Siebert and Veeser, Theory of adaptive finite el-
ement methods: an introduction
Multiscale, Nonlinear and Adaptive Approximation
Springer, 2009
• Bonito and Nochetto, Quasi-optimal convergence rate of
an adaptive discontinuous Galerkin method
SIAM J. Numer. Anal., 2010