Global Approximation of Singular Capillary Surfaces ...
Transcript of Global Approximation of Singular Capillary Surfaces ...
Global Approximation of Singular Capillary Surfaces: asymptotic analysis meets numerical analysis
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Yasunori Aoki Hans De Sterck
Liquid Surface at a corner
Liquid Surface at a corner
Liquid Surface at a corner
Liquid Surface at a corner
Liquid Surface at a corner
Liquid Surface at a corner
Height of the capillary surface depends discontinuously on the wedge angle.
(Paul Concus and Robert Finn)
Laplace-Young Equation
In a domain with a sharp corner, the solution becomes unbounded. (Concus and Finn)
�
� <⇤
2� ⇥
⌦ = {(r, ✓) : 0 < r < R , �↵ < ✓ < ↵}
Wedge
�
� <⇤
2� ⇥
⌦ = {(r, ✓) : 0 < r < R , �↵ < ✓ < ↵}
Wedge
�
� <⇤
2� ⇥
⌦ = {(x, y) : 0 < x, f2(x) < y < f1(x)}
Cusp
�1 + �2 6= ⇡
Asymptotic Analysis
Approximating Laplace-Young Equation
|�u| >> 1
� · �u
|�u| � u
� · �u�1 + |�u|2
= u in �
Asymptotic Laplace-Young Equation(two lucky cases)
� · �v
|�v|2 = cos � on ��
� · �v
|�v|2 = v in �
Asymptotic Laplace-Young Equation(two lucky cases)
� · �v
|�v|2 = cos � on ��
� · �v
|�v|2 = v in �
v(r, �) =cos � �
�k2 � sin2 �
kr
(Concus and Finn, Miersemann, and King et al.)
Asymptotic Laplace-Young Equation(two lucky cases)
� · �v
|�v|2 = cos � on ��
� · �v
|�v|2 = v in �
u(r, �) = v(r, �) + O(r3) as r � 0
(Miersemann)
v(r, �) =cos � �
�k2 � sin2 �
kr
(Concus and Finn, Miersemann, and King et al.)
Asymptotic Laplace-Young Equation(two lucky cases)
�
� · �v
|�v|2 = cos � on ��
� · �v
|�v|2 = v in �
Asymptotic Laplace-Young Equation(two lucky cases)
�
� · �v
|�v|2 = cos � on ��
� · �v
|�v|2 = v in �
Asymptotic Laplace-Young Equation(two lucky cases)
�
� · �v
|�v|2 = cos � on ��
� · �v
|�v|2 = v in �
u(p, q) = v(p, q) + O(p�5) as p��
(Aoki M.Math thesis)
Asymptotic Analysis(general cases)
�1 + �2 6= ⇡
after some calculation ...
Asymptotic Analysis(general cases)
�1 + �2 6= ⇡
after some calculation ...
u(x, y) =cos �1 + cos �2
f1(x)� f2(x)+ O
�f �1(x)� f �
2(x)f1(x)� f2(x)
�as x� 0+
* some restrictions on and applyf1 f2
(Aoki and Siegel)
Asymptotic Analysis(summary)
u(x, y) � cos �1 + cos �2
f1(x)� f2(x)
u(r, �) � cos � ��
k2 � sin2 �
krCorner:
Cusp:
Approximation only accurate near the singularity!
Numerical Analysis
�1 �2
�3�4
�5
�i
Finite Element ApproximationBasis Functions
u � uh :=Nnode�
i=1
ci�i
Standard Trial Function Expansion
Finite Element Approximation(Function Spaces)
Function Spaces in Bounded Lipschitz Domain
L1L2
W1,1
H1
uh
Bounded Solutions of the Laplace-Young EquationUnbounded Solutions of the Laplace-Young Equation
Asymptotic Analysis
Corner: f1(x)
f2(x)
u = O
�1r
�= O
�1
f1(x)� f2(x)
�
=O(1)
f1(x)� f2(x)
Asymptotic Analysis
Corner: f1(x)
f2(x)
u = O
�1r
�= O
�1
f1(x)� f2(x)
�
=O(1)
f1(x)� f2(x)
u = O
�1
f1(x)� f2(x)
�Cusp:
�1 + �2 6= ⇡
=O(1)
f1(x)� f2(x)
Asymptotic Analysis
u =O(1)
f1(x)� f2(x)
Asymptotic Analysis
u =O(1)
f1(x)� f2(x)
Change of Variable
u =v
f1(x)� f2(x)
Bounded function
Unbounded function
Asymptotic Analysis
u =O(1)
f1(x)� f2(x)
Change of Variable
u =v
f1(x)� f2(x)
u ��Nnode
i=1 ci�i
f1(x)� f2(x)
Change of Coordinates
f1(x)
f2(x)
10 x
y
t =2y � (f1 + f2)
f1 � f2
s = x
x = s
y =1 + t
2f1(s) +
1� t
2f2(s)
s
t1
1
�1
0
Change of Coordinates
f1(x)
f2(x)
10 x
y
s
t1
1
�1
0
Finite Volume Element method
u � C2(�)
� · �u�1 + |�u|2
= u
Let be the solution of the following PDE:
�
��
� · �u�1 + |�u|2
dA =�
��
u dA for all �� � �
�
���
� · �u�1 + |�u|2
ds =�
��
u dA for all �� � �
Finite Volume Element method
�1
�2
Finite Volume Element method
�
���
� · �u�1 + |�u|2
ds =�
��
u dA � · �u�1 + |�u|2
= cos � on ��+
Finite Volume Element method
�
���
� · �u�1 + |�u|2
ds =�
��
u dA � · �u�1 + |�u|2
= cos � on ��+
�
���\��� · �u�
1 + |�u|2ds +
�
������cos �ds =
�
��
u dA
for all �� � �
Finite Volume Element method
�
���
� · �u�1 + |�u|2
ds =�
��
u dA � · �u�1 + |�u|2
= cos � on ��+
�
���\��� · �u�
1 + |�u|2ds +
�
������cos �ds =
�
��
u dA
for all �� � �
u ��Nnode
i=1 ci�i
f1(x)� f2(x)
We now approximate the solution with a finite element approximation
= uh
Finite Volume Element method
�
���\��� · �u�
1 + |�u|2ds +
�
������cos �ds =
�
��
u dA
for all �� � �
Finite Volume Element method
�
���\��� · �u�
1 + |�u|2ds +
�
������cos �ds =
�
��
u dA
for all �� � �
for j = 1, 2, . . . , Nnode
�
��j\��� ·
�Nnodei=1 ci�
��i
f1(x)�f2(x)
�
�1 + |
�Nnodei=1 ci�
��i
f1(x)�f2(x)
�|2
ds +�
��j���cos �ds
=�
�j
Nnode�
i=1
ci
��i
f1(x)� f2(x)
�dA
Finite Volume Element method
Finite Volume Method Control Volumes
�1
�2
�3�4
�5�i
Finite Element Expansion
Ω5
Ω4
Ω3
Ω2
Ω1
Ω10
Ω9
Ω8
Ω7
Ω6
Ω15
Ω14
Ω13
Ω12
Ω11
Ω20
Ω19
Ω18
Ω17
Ω16
Ω25
Ω24
Ω23
Ω22Ω21
+
for j = 1, 2, . . . , Nnode
�
��j\��� ·
�Nnodei=1 ci�
��i
f1(x)�f2(x)
�
�1 + |
�Nnodei=1 ci�
��i
f1(x)�f2(x)
�|2
ds +�
��j���cos �ds
=�
�j
Nnode�
i=1
ci
��i
f1(x)� f2(x)
�dA
Finite Volume Element method
f1(x)
f2(x)
10 x
y
s
t1
1
�1
0
t =2y � (f1 + f2)
f1 � f2
s = x
x = s
y =1 + t
2f1(s) +
1� t
2f2(s)
for j = 1, 2, . . . , Nnode
�
��j\��� ·
�Nnodei=1 ci�
��i
f1(x)�f2(x)
�
�1 + |
�Nnodei=1 ci�
��i
f1(x)�f2(x)
�|2
ds +�
��j���cos �ds
=�
�j
Nnode�
i=1
ci
��i
f1(x)� f2(x)
�dA
�
�j
· dA
� �
�j
· f1(s)� f2(s)2
ds dt
Finite Volume Element method
f1(x)
f2(x)
10 x
y
s
t1
1
�1
0
t =2y � (f1 + f2)
f1 � f2
s = x
x = s
y =1 + t
2f1(s) +
1� t
2f2(s)
for j = 1, 2, . . . , Nnode
�
��j\��� ·
�Nnodei=1 ci�
��i
f1(x)�f2(x)
�
�1 + |
�Nnodei=1 ci�
��i
f1(x)�f2(x)
�|2
ds +�
��j���cos �ds
�
�j
· dA
� �
�j
· f1(s)� f2(s)2
ds dt
=12
� �
�j
Nnode�
i=1
ci�i ds dt
Finite Volume Element method
Numerical Experiments
Asymptotic Laplace-Young Equation(in a Corner domain)
� · �v
|�v|2 = cos � on ��
� · �v
|�v|2 = v in �
v(r, �) =cos � �
�k2 � sin2 �
kr
u(r, �) = v(r, �) + O(r3) as r � 0
WithoutChange of Variable
WithChange of Variable
Regular Coordinates (Scott et al.)
Curvilinear Coordinates
Convergence Study(Asymptotic Laplace-Young Equation in a Corner domain)
L1 E
rror
Number of unknowns
1E+00 1E+01 1E+02 1E+03 1E+041E-05
1E-04
1E-03
1E-02
1E-01
1E+00
Regular Trial Function + Regular CoordinateAsymptotic Anslysis inspired Trial Function + Regular CoordinateRegular Trial Function + Curvilinear CoordinateAsymptotic Anslysis inspired Trial Function + Curvilinear Coordinate
Convergence Study(Asymptotic Laplace-Young Equation in a Corner domain)
Without Change of Variable
WithChange of Variable
Regular Coordinates Linear Linear
Curvilinear Coordinates Linear Quadratic
Convergence Study(Asymptotic Laplace-Young Equation in a Corner domain)
Asymptotic Laplace-Young Equation(in a Circular Cusp domain)
�
� · �v
|�v|2 = cos � on ��
� · �v
|�v|2 = v in �
u(p, q) = v(p, q) + O(p�5) as p��
1E+00 1E+01 1E+02 1E+03 1E+041E-06
1E-05
1E-04
1E-03
1E-02
1E-01
1E+00
Number of Unknowns ( )
L1 E
rror
Convergence Study(Asymptotic Laplace-Young Equation in a Circular Cusp domain)
FEM with change of coordinates and without change of variable FEM with change of coordinates and with change of variableFVEM with change of coordinates and without change of variable FVEM with change of coordinates and with change of variable
1E+00 1E+01 1E+02 1E+03 1E+040
1
2
3
4
Number of Unknowns ( )
Con
verg
ence
Ord
er
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xy
u(x
,y)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
� = �/6
�=
0
� = �/6
Numerical Experiment(Laplace Young Equation in a Corner domain)
y
x
Numerical Experiment(Finite Volume Element approximation with change of variable and with change of coordinates)
t
−1 −0.5 0 0.5 11
1.5
2
2.5
3
3.5
Numerical Experiment(Finite Volume Element approximation with change of variable and with change of coordinates)
t
−1 −0.5 0 0.5 11
1.5
2
2.5
3
3.5
Analytically Proven Asymptotic Behaviour
cos � ��
k2 � sin2 �
kr
�����s=h
Numerical Solution
uh(s, t)|s=h
t
−1 −0.5 0 0.5 15.5
6
6.5
7
Numerical Experiment(Finite Volume Element approximation with change of variable and with change of coordinates)
t
−1 −0.5 0 0.5 111.5
12
12.5
13
13.5
14
Numerical Experiment(Finite Volume Element approximation with change of variable and with change of coordinates)
t
−1 −0.5 0 0.5 123
23.5
24
24.5
25
25.5
26
26.5
27
27.5
28
Numerical Experiment(Finite Volume Element approximation with change of variable and with change of coordinates)
� = 0
� = �/6
� = �/4
t−1 −0.5 0 0.5 1
10
15
20
25
30
35
40
45
50
Numerical Experiment(Finite Volume Element approximation with change of variable and with change of coordinates)
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0
0.5
0
5
10
15
20
25
30
35
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xy
u(x
,y)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
� = �/6
�=
0
� = �/6
Numerical Experiment(Laplace Young Equation in a Corner domain)
y
x
10−2 10−1 10010−1
100
101
102
Leading order term of the asymptotic solutionNumerical solution
−2 0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
10
u(s
,0)
s
MP 1-1MP 1-2
MP 1-3
Numerical Experiment(Finite Volume Element approximation with change of variable and with change of coordinates)
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1000
2000
3000
4000
5000
6000
7000
8000 0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5
2
�=
�/6
�=
0
� = �/6
Numerical Experiment(Laplace-Young Equation in a Cusp Domain)
y
x
10−2 10−1 100100
101
102
103
104
105
106
107
Asymptotic ApproximationNumerical Solution
−2 0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
10
Numerical Experiment(Finite Volume Element approximation with change of variable and with change of coordinates)
Open Problems
Theorem 2: is bounded if , and .
u cos �1 + cos �2 = 0f1,2,(x) � C2([0,�))
Y.A. and David Siegel, Bounded and Unbounded Capillary Surfaces in a Cusp Domain, to appear in Pacific Journal of Mathematics, accepted on December 31st 2011.
Open Problem 1: Is bounded if , and .
u cos �1 + cos �2 = 0
f1,2(x) /� C2([0,�))
f1(x) =16x
32 f2(x) =
18x
32For example
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Open Problem 1: Is bounded if , and .
u cos �1 + cos �2 = 0
f1,2(x) /� C2([0,�))
f2(x) =18x
32f1(x) =
16x
32
�1 =56� +
�
180�1 =
56� � �
180
�2 =16� �2 =
16�
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Open Problem 1: Is bounded if , and .
u cos �1 + cos �2 = 0
f1,2(x) /� C2([0,�))
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�1 =56�
�2 =16�
f2(x) =18x
32f1(x) =
16x
32
The formal asymptotic series solution for the regular cusp domain:
v =cos �1 + cos �2
f1(x)� f2(x)+ g(x, y)
f �1(x)� f �
2(x)f1(x)� f2(x)
+ h(x, y)(f �
1(x)� f �2(x))2
f1(x)� f2(x)
g(x, y) = �
�
1��
cos �1(t + 1) + cos �2(t� 1)2
�2
h(x, y) = �cos �1 + cos �2
4
��t +
t2
2
�+
1� �
2(cos �1 + cos �2g(x, y)2
t =2y � (f1(x) + f2(x))
f1(x)� f2(x)
Open Problem 2:
What is the formal asymptotic series for the double cusp domain?
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−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−100
−90
−80
−70
−60
−50
−40
Asymptotic Approximation of the second order termNumerical Solution of the second order term
−2 0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
10
f1(x) =16x3f2(x) = �1
8x3
t
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−100
−50
0
50
100
150
Numerical Solution of the second order term
−2 0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
10
�
�
1��
cos �1(t + 1) + cos �2(t� 1)2
�2 f �1(x)� f �
2(x)f1(x)� f2(x)
The formal asymptotic series solution for the double cusp domain:
t =2y � (f1(x) + f2(x))
f1(x)� f2(x)
Open Problem 2:
What is the formal asymptotic series for the double cusp domain?
v =cos �1 + cos �2
f1(x)� f2(x)+ g(x, y)
f �1(x)� f �
2(x)f1(x)� f2(x)
+ h(x, y)(f �
1(x)� f �2(x))2
f1(x)� f2(x)
g(x, y) = �
�
1��
cos �1(t + 1) + cos �2(t� 1)2
�2
h(x, y) =?
+C
Conclusion
Asymptotic Analysis
Asymptotic Analysis
Change of Variable
Asymptotic Analysis
Change of Variable + Curvilinear Coordinate System
Asymptotic Analysis
Change of Variable + Curvilinear Coordinate System
Finite Volume Element methodor
Finite Element method
Asymptotic Analysis
Change of Variable + Curvilinear Coordinate System
Numerical Approximation valid for the entire domain
Finite Volume Element methodor
Finite Element method