Numerical approximation of homogeneous
Poincaré - Friedrichs constants
Lyonell Boulton (Heriot-Watt University)
Third Scottish PDEs Colloquium
1-2 June 2015
The main goal of this talk
Compute kΣ > 0
∫
∫
| grad u|2
|u| ≤ kΣ
2
Σ
∀u ∈ H01 (Σ)
Σ
for ∂Σ a Koch snowflake
Challenge:
kΣ as small as possible
inspiration:
Tomáš Vejchodský’s seminar University of Strathclyde March 2014
1
0.5
∂Σ
=
0
-0.5
-1
-1
-0.5
0
0.5
1
1
0.5
∂Σ
=
0
-0.5
-1
-1
-0.5
0
0.5
1
Our conjecture so far
optimal kΣ <
10000
130365
The Method
Main steps
1. Embed Σ into a family of slightly larger polyhedra Ωj and
recall domain monotonicity. We call these larger domains
the levels.
2. Find conformal maps from unit square S into Ωj .
3. Write the corresponding eigenvalue problem in S for level
as system of order 1.
4. Compute certified lower bound for smallest positive
eigenvalue of system via the quadratic method.
Steps 1
Domain Monotonicity
1
0.5
⇒
0
-0.5
-1
-1
Σ ⊂ D = {x2 + y 2 ≤ 1}
-0.5
0
0.5
1
optimal kΣ ≤
1
10000
<
(j0,1 )2
57830
Domain Monotonicity
1
0.5
⇒
0
optimal kΣ ≤
1
10000
<
(j0,1 )2
57830
-0.5
-1
-1
Σ ⊂ D = {x2 + y 2 ≤ 1}
-0.5
compare:
0
0.5
10000
57830
1
≈ 0.1729
vs
10000
130365
≈ 0.0767
Level 0
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Level 0 to 1
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Level 1
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Level 2
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Level 3
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Level 4
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Σ ⊂ Ωj
⇒
optimal kΣ ≤ optimal kΩj
Σ ⊂ Ωj
⇒
optimal kΣ ≤ optimal kΩj
We have found so far
optimal kΩ4 ≤
10000
130365
Steps 2
The comformal maps
idea from [Banjai 2007]
1
0
0.5
-0.2
f − conformal
-0.4
0
S −→ Ω
-0.6
-0.8
-0.5
-1
-1
0
{
0.2
0.4
0.6
0.8
−∆v = ω 2 |f ′ |2 v
v=0
1
-1
in S
on ∂S
v =u◦f
{
-0.5
0
−∆u = ω 2 u
u=0
0.5
1
in Ω
on ∂Ω
Schwarz-Christoffel formula
∫
fj (z) = fj (z0 ) + C
z n(j)
∏(
z0 k=1
ζ
1−
zk
)αk −1
n(j) = 3j+2 − 3
zk ∈ ∂S
pre-vertices
f (zk ) = wk
wk ∈ ∂Ωj
all the corners
{
interior angles are παk =
2π/3
5π/3
dζ
Steps 3
The system formulation
In Ω ≡ Ωj
z

G
0
i div
i grad
0
{ z
dom G
L2 (Ω)3
}| {
z }| {
0
i div
H01 (Ω)
L2 (Ω)

:
×
−→ ×
2
H(div, Ω)
i grad
0
L2 (Ω)2
[
}|
][ ]
[ ]
u
u
=ω
σ
σ
⇐⇒
ω2u =
− div grad
| {z }
−∆:H01 ∩H 2 −→L2
u
The system formulation
In Ω ≡ Ωj
z

G
0
i div
i grad
0
{ z
L2 (Ω)3
dom G
}| {
z }| {
0
i div
H01 (Ω)
L2 (Ω)

:
×
−→ ×
2
H(div, Ω)
i grad
0
L2 (Ω)2
[
}|
][ ]
[ ]
u
u
=ω
σ
σ
⇐⇒
ω2u =
− div grad
| {z }
−∆:H01 ∩H 2 −→L2
1
optimal kΩ = 2
ωg
∫
ωg2
= min
u∈H01
| grad u|2
Ω∫
2
Ω |u|
u
In S
′ 2
−∆v = ω |f | v
2
⇐⇒
ω ̸= 0
2
[ ′
]
|f | 0
F =
0 I
[ ]
[ ]
v
2 v
G
= ωF
s
s
In S
′ 2
−∆v = ω |f | v
2
⇐⇒
ω ̸= 0
2
[ ]
[ ]
v
2 v
G
= ωF
s
s
[ ′
]
|f | 0
F =
0 I
that is
[ ]
[ ]
w
−1
−1 w
F
GF
=
ω
| {z } t
t
T
for
[ ]
[ ]
w
v
=F
t
s
In S
′ 2
−∆v = ω |f | v
2
⇐⇒
ω ̸= 0
2
[ ]
[ ]
v
2 v
G
= ωF
s
s
[ ′
]
|f | 0
F =
0 I
that is
[ ]
[ ]
w
−1
−1 w
F
GF
=
ω
| {z } t
t
T
for
[ ]
[ ]
w
v
=F
t
s
we find lower bounds for the first non-zero positive
eigenvalue of the densely defined self-adjoint operator T
which acts in L2 (S)
Steps 4
The quadratic method
finite-dimensional
Find λ ∈ C and 0 ̸=
⟨
L ⊂ dom T
[ ]
w
∈ L such that
t
[ ]
[ ]⟩
w
w̃
(T − λ)
, (T − λ)
=0
t
t̃
[ ]
w̃
∀
∈L
t̃
The quadratic method
finite-dimensional
Find λ ∈ C and 0 ̸=
⟨
⇐⇒
L ⊂ dom T
[ ]
w
∈ L such that
t
[ ]
[ ]⟩
w
w̃
(T − λ)
, (T − λ)
=0
t
t̃
[ ]
w
Q(λ)
=0
t
[ ]
w̃
∀
∈L
t̃
Q(z) = K − 2zL + z 2 M
The quadratic method
L ⊂ dom T
finite-dimensional
Find λ ∈ C and 0 ̸=
⟨
⇐⇒
[ ]
w
∈ L such that
t
[ ]
[ ]⟩
w
w̃
(T − λ)
, (T − λ)
=0
t
t̃
[ ]
w
Q(λ)
=0
t
K = [⟨T bj , T bk ⟩]djk=1
L=
[ ]
w̃
∀
∈L
t̃
Q(z) = K − 2zL + z 2 M
L = [⟨T bj , bk ⟩]djk=1
span{bj }dj=1
M = [⟨bj , bk ⟩]djk=1
[ ] ∑
d
w
=
uj bj
t
j=1
[Davies 1998], [Shargorodsky 2000], [Levitin-Shargorodsky
2004], [Boulton-Strauss 2011]
{
}
a + b b − a
D(a, b) = z ∈ C : z −
<
2 2
[Shargorodsky 2000]
(a, b) ∩ spec T = ∅
⇒
det Q(λ) ̸= 0 ∀λ ∈ D(a, b)
Spectral inclusions
{
}
a + b b − a
<
D(a, b) = z ∈ C : z −
2 2
[Boulton-Levitin 2006 / Strauss 2007]

(a, b) ∩ spec T = {ω}

| Im λ|2
| Im λ|2
det Q(λ) = 0
⇒ Re λ−
< ω < Re λ+

b − Re λ
Re λ − a

λ ∈ D(a, b)
Ω ⊂ Ω̃ ⇒ ω22 (Ω̃) ≤ ω22 (Ω)
Therefore take a = 0 and b ≤ ω2 (Ω̃) known for suitable Ω̃
det Q(λ) = 0
}
λ ∈ D(0, b)
⇒ Re λ −
pick
Ω̃ = D
⇒
b=
√
| Im λ|2
< ωg (Ω)
b − Re λ
14.68 < j1,1
is fine
In triangulations Th of S pick
[
}
{[ ]
]
v|K
v
3
3
0
L̂ =
∈ C (S) :
∈ Pp (K) ∀K ∈ Th , v|∂S = 0
σ|K
σ
In triangulations Th of S pick
[
}
{[ ]
]
v|K
v
3
3
0
L̂ =
∈ C (S) :
∈ Pp (K) ∀K ∈ Th , v|∂S = 0
σ|K
σ
[ ]
[ ]
v
ṽ
Find λ and 0 ̸=
∈ L̂ such that for all
∈ L̂
σ
σ̃
⟨
[ ]
[ ]⟩
⟨ [ ] [ ]⟩
⟨ [ ] [ ]⟩
v
ṽ
v
ṽ
v
ṽ
F −1 G
, F −1 G
−2λ G
,
+λ2 F
,F
=0
σ
σ̃
σ
σ̃
σ
σ̃
[
0
i div
G=
i grad
0
]
[
|f ′ | 0
F =
0 I
]
Level
0
1
2
3
4
(ω12 )upper
lower
7.1553397
80
11.78146
35
12.51993
37
12.8981
73
13.0378
65
NDOFS
21753
39378
109473
109473
109473
hmax
0.07
0.05
0.03
0.03
0.03
p
5
5
5
5
5
Software:
Driscoll’s Schwarz-Christoffel Toolbox for Matlab + Comsol LiveLink
Conclusions
▶
Method is brilliant
▶
No need for “convergence analysis” one iteration gives
upper (and lower) bound for the gradient
Poincaré-Friedrichs constant
▶
Similar approached (sans domain monotonicity) can be
applied to curl: can prove convergence but so far no
one-step certification
...more precise references upon request
Many thanks!