global approximation of singular capillary surfaces

Global Approximation of Singular Capillary Surfaces: asymptotic analysis meets numerical analysis

00.2

0.40.6

0.81 −1

−0.50

0.512

4

6

8

10

12

14

Yasunori Aoki Hans De Sterck

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 �


� · �v

|�v|2 = cos � on ��

� · �v

|�v|2 = v in �

v(r, �) =cos � �

�k2 � sin2 �

kr

(Concus and Finn, Miersemann, and King et al.)


� · �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.)


�

� · �v

|�v|2 = cos � on ��

� · �v

|�v|2 = v in �


�

� · �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 ��


�1

�2


�

��

� · �u�1 + |�u|2

ds =�

��

u dA � · �u�1 + |�u|2

= cos � on ��+


�

��

� · �u�1 + |�u|2

ds =�

��

u dA � · �u�1 + |�u|2

= cos � on ��+

�

��\�� · �u�

1 + |�u|2ds +

�

��cos �ds =

�

��

u dA

for all ��


�

��

� · �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


�

��\�� · �u�

1 + |�u|2ds +

�

��cos �ds =

�

��

u dA

for all ��


�

��\�� · �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 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

Nnode�

i=1

ci

��i

f1(x)� f2(x)

�dA


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

Nnode�

i=1

ci

��i

f1(x)� f2(x)

�dA

�

�j

· dA

� �

�j

· f1(s)� f2(s)2

ds dt


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

· dA

� �

�j

· f1(s)� f2(s)2

ds dt

=12

� �

�j

Nnode�

i=1

ci�i ds dt


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


Without Change of Variable

WithChange of Variable

Regular Coordinates Linear Linear

Curvilinear Coordinates Linear Quadratic


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

00.2

0.40.6

0.81 −0.5

0

0.5

0

5

10

15

20

25

30

35

40

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


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


t

−1 −0.5 0 0.5 111.5

12

12.5

13

13.5

14


t

−1 −0.5 0 0.5 123

23.5

24

24.5

25

25.5

26

26.5

27

27.5

28


� = 0

� = �/6

� = �/4

t−1 −0.5 0 0.5 1

10

15

20

25

30

35

40

45

50


00.2

0.40.6

0.81 −0.5

0

0.5

0

5

10

15

20

25

30

35

40

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


00.2

0.40.6

0.81 −0.2

−0.10

0.10.2

0

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


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

00.2

0.40.6

0.81 0

0.050.1

0.150.2

0

50

100

150

200

250

00.2

0.40.6

0.81 0

0.050.1

0.150.2

−250

−200

−150

−100

−50

0

50


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�

00.2

0.40.6

0.81 0

0.050.1

0.150.2

0

50

100

150

200

250

00.2

0.40.6

0.81 0

0.050.1

0.150.2

−250

−200

−150

−100

−50

0

50


u cos �1 + cos �2 = 0

f1,2(x) /� C2([0,�))

00.2

0.40.6

0.81 0

0.050.1

0.150.2

0.2

0.25

0.3

0.35

0.4

0.45

0.5

�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?

00.2

0.40.6

0.81 −0.2

−0.10

0.10.2

0

1000

2000

3000

4000

5000

6000

7000

8000

−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


Finite Volume Element methodor

Finite Element method

Asymptotic Analysis


Numerical Approximation valid for the entire domain

Finite Volume Element methodor

Finite Element method

global approximation of singular capillary surfaces

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