Multiple Solutions of the Lane-Emden Problem


Results obtained by G. Chen, A. Perronnet, J. Zhou






Some 2d numerical results for the Lane-Emden problem with p=3:

The domain is composed of 2 disks and a corridor.
The mesh has 6138 triangles, 3226 vertices, 12589 nodes (vertices and edge middles).
The interpolation uses polynomials of degree 2.

Boundary condition of Dirichlet:

The results are written in the order of ascending energy level E.

The initial value u0(x) is 1 on the right disk and 0 otherwise. sM=(2,0)
After 12 iterations, the final values are between 0 and 3.56 reached at (1.98017, 0.000418)
The energy value is 10.9049 . Epsilon=3x10-4
This is the ground state of the Lane-Emden equation which is a single-peak solution concentrated mainly on the large disk.

The initial value u0(x)=cos(Π/2|x-(-1,0)|) on the left disk and 0 otherwise. sM=(-1,0)
After 57 iterations, the final values are between 0 and 7.012 reached at (-0.970226, -0.0066)
The energy value is 42.8252 . Epsilon=6x10-3
The positive single-peak solution is mainly concentrated on the small disk.


The initial value u0(x)=0 inside the 2 disks, 1 otherwise. SM=(0,0)
After 100 iterations (non convergence), the final values are between 0 and 13.36 reached at (0.2319, 0.000135)
The energy value is 165.82 . Epsilon=6x10-3
The positive single-peak solution is mainly concentrated on the corridor.

Boundary condition of Neumann:

This Lane-Emden problem may be obtained by a modelisation of a stationary system of chemotactic aggregation of cellular slime molds, with a changement of variables. The Laplacian is preceded of a small positive coefficient alpha. Multiple solutions exist according to the variation of curvature of the boundary (Cf. Ni and Takagi, Gui and Wei, ...)

The followins results are obtained with a coefficient alpha value equals to 1.
The first spike-layer solution, from u0=1 on the domain, is the trivial solution, equal to 1 everywhere, obtained in 1 iteration, and not visualized here.
The ground state spike-layer solution is obtained from the initial value u0(x)=1 inside the left disk, 0 otherwise. SM=(-1,0)
After 10 iterations, the final values are between 0 and 1.22637 reached at (-1.4998, -0.0125) on the boundary
This is consistent with the theory (Cf. Ni and Takagi) that the maximum appeares at a point with the maximum curvature.
The energy value is 0.373828 . Epsilon=10-3


The second positive spike-layer solution is obtained from the initial value u0(x)=1 inside the right disk, 0 otherwise. SM=(2,0)
After 32 iterations, the final values are between 0 and 1.1727 reached at (3,0) on the boundary
The energy value is 0.919049 . Epsilon=10-3

It has been impossible to obtain a solution only within the corridor. From a non zero initialisation of u0(x) inside the corridor, the solution is always attracted by the ground state solution which seems more stable than the second solution. This suggests that there are no spike-layer only inside the corridor region.

Boundary condition of Robin:

The initial value u0(x) equal to 1 on the right disk and 0 otherwise. sM=(2,0)
After 9 iterations, the final values are between 0 and 1.60458 reached at (2.10118, -0.00273)
The energy value is 1.95854 . Epsilon=10-3
The ground state single spike-layer solution is concentrated mainly on the right disk and the corridor.

The initial value u0(x) equal to 1 on the left disk and 0 otherwise. sM=(-1,0)
After 7 iterations, the final values are between 0 and 2.258 reached at (-1.09699, -0.00317)
The energy value is 2.91239 . Epsilon=10-3
The single spike-layer solution is concentrated mainly on the left disk and the corridor.

The initial value u0(x) equal to 0 on the 2 disks and 1 otherwise. sM=(0,0)
After 52 iterations, the final values are between 0 and 3.228 reached at (0.16846, -0.005589)
The energy value is 5.58789 . Epsilon=10-3
The single spike-layer solution is concentrated mainly on the corridor.

The Lane-Emden problem with a Robin boundary condition on a nonconvex, nonstar-shaped, multiply connected 2d domain:

A coefficient alpha is put also before the Laplacian and the problem is to obtain u(x) solution of

α=10-3, β=1   the heat transfer coefficient 

The domain is 
- a disk centered at (-1,0) with a radius of 0.5;
- a great ellipse centered at (2,0) with a X-axis lenght of 0.5 and Y-axis lenght of 1.0;
- a small ellipse centered at (2,0) with a X-axis lenght of 0.1 and Y-axis lenght of 0.5;
- a corridor between the balls with a radius of 0.2;
The mesh has 6347 vertices, 24724 nodes or degrees of freedom (dof), 12030 triangles.




The initial conditions around the point (2.3,0) are defined by a user's function:
DefFunc temperature0( t, x, y, z, nty, nuob );
  if  2.1
Results: CONVERGENCE after  21 ITERATIONS
 MAX|U(N)|=   2.16577       At NODE XYZ=( 2.29057, -0.850032E-01 )
 ||Un||   =  0.598213E-01   ||Un-Un-1||/||Un||=  0.479523E-03
 | Un |H1 =  0.153415       | Hn-Hn-1 |/| Hn |=  0.463385E-03
 J=ENERGY =  0.588403E-02   ||En-En-1||/||En||=  0.926986E-03
 |RESIDU0|=  0.948523E-07   Max|RESIDUE Un(P)|=  0.948281E-07
Scaling Factor = (1/U)**(1/(P-1))= 0.131113278 





The initial conditions around the point (2,0.75):
DefFunc temperature0( t, x, y, z, nty, nuob );
  if  y>0.5 et abs(x-xc2)<0.2  then
    temperature0 = 1.0;
  else
    temperature0 = 0.0;
  endif;
EndFunc;
At point (2,0.75)  temperature0=1.00001

Results: CONVERGENCE after 154 ITERATIONS
 MAX|U(N)|=   2.15873       At NODE XYZ=( 2.03546, 0.747884 )
 ||Un||   =  0.600030E-01   ||Un-Un-1||/||Un||=   0.00000    
 | Un |H1 =  0.152818       | Hn-Hn-1 |/| Hn |=   0.00000    
 J=ENERGY =  0.587512E-02   ||En-En-1||/||En||=   0.00000    
 |RESIDU0|=  0.667051E-05   Max|RESIDUE Un(P)|=  0.667051E-05
Scaling Factor = (1/U)**(1/(P-1))=  1. 





The initial conditions around the point (1.71,0):
DefFunc temperature0( t, x, y, z, nty, nuob );
DefVar r; r=sqrt((x-1.71)**2+y**2);
  if  r<0.3  then
    temperature0 = 1.0;
  else
    temperature0 = 0.0;
  endif;
EndFunc;
At point (1.71,0)  temperature0=1.00001

Results: CONVERGENCE after 138 ITERATIONS
 MAX|U(N)|=   2.16453       At NODE XYZ=( 1.65689, 0.294089E-02 )
 ||Un||   =  0.612721E-01   ||Un-Un-1||/||Un||=  0.998909E-03
 | Un |H1 =  0.153463       | Hn-Hn-1 |/| Hn |=  0.154950E-04
 J=ENERGY =  0.588775E-02   ||En-En-1||/||En||=  0.309892E-04
 |RESIDU0|=  0.643943E-05   Max|RESIDUE Un(P)|=  0.643774E-05
Scaling Factor = (1/U)**(1/(P-1))= 0.425304024





The initial conditions around the point (0.31,0):
DefFunc temperature0( t, x, y, z, nty, nuob );
  if  -0.51+0.05
Results: CONVERGENCE after 53 ITERATIONS
 MAX|U(N)|=   2.18230       At NODE XYZ=( 0.270527,  0.214300E-02 )
 ||Un||   =  0.597973E-01   ||Un-Un-1||/||Un||=  0.679739E-03
 | Un |H1 =  0.153316       | Hn-Hn-1 |/| Hn |=  0.489756E-03
 J=ENERGY =  0.587646E-02   ||En-En-1||/||En||=  0.979393E-03
 |RESIDU0|=  0.222410E-04   Max|RESIDUE Un(P)|=  0.222012E-04
Scaling Factor = (1/U)**(1/(P-1))= 0.74283237





The initial conditions around the point (-0.953,0):
DefFunc temperature0( t, x, y, z, nty, nuob );
  if  -0.953-0.5
Results: CONVERGENCE after 21 ITERATIONS
 MAX|U(N)|=   2.16432       At NODE XYZ=( -0.989058,  0.499843E-02 )
 ||Un||   =  0.641153E-01   ||Un-Un-1||/||Un||=   0.00000    
 | Un |H1 =  0.153614       | Hn-Hn-1 |/| Hn |=   0.00000    
 J=ENERGY =  0.588185E-02   ||En-En-1||/||En||=   0.00000    
 |RESIDU0|=  0.175929E-04   Max|RESIDUE Un(P)|=  0.175929E-04
Scaling Factor = (1/U)**(1/(P-1))=  1.


Some 3d numerical results for the Lane-Emden problem with p=3:

The domain is composed of 3 balls and a corridor.
The mesh has 15997 tetrahedra, 3061 vertices, 22833 nodes (vertices and edge middles).
The interpolation uses polynomials of degree 2.

Boundary condition of Dirichlet with p=3, a=0:

The initial value u0(x) is 1 on the right ball and 0 otherwise. sM=(0,2,0)
After 32 iterations, the final values are between 0 and 7.868 reached at the point +M of the graphic
The energy value is 29.82 . Epsilon=10-3
This is the ground state spike-layer of the Lane-Emden equation which is a solution concentrated mainly into the large ball under the small ball hole.

The initial value u0(x) is 1 on the left ball and 0 otherwise. sM=(0,-1,0)
After 59 iterations, the final values are between 0 and 13.93 reached at the point +M of the graphic
The energy value is 52.79 . Epsilon=10-3
This solution is concentrated mainly into the left ball.

The initial value u0(x) is 1 on the corridor and 0 otherwise. sM=(0,-1,0)
After 100 iterations the solution is in fact the ground state!
Various attempts by choosing different u0(x) and sM for iterations result either in divergence of iterates or in convergence of iterations to the ground state.
It seems that here there are no solution concentrated into the corridor.

Boundary condition of Neumann with p=3, a=1:

The initial value u0(x) is 1 on the left ball and 0 otherwise. sM=(0,-1,0)
After 13 iterations, the final values are between 0 and 1.114 reached at the point +M of the graphic
The energy value is 0.1818 . Epsilon=2x10-3
The maximum value +M of this solution is reached on the boundary, consistent with Ni and Takagi.

The initial value u0(x) is 1 on the right ball and 0 otherwise. sM=(0,2,0)
After 6 iterations, the final values are between 0 and 1.034 reached at the point +M of the graphic
The energy value is 1.059 . Epsilon=2x10-3


As for the 2d case, it is not expected the existence of a spike-layer solution whose spike occurs on the corridor region.

Boundary condition of Robin p=3, a=0, b=1:

The initial value u0(x) is 1 on the left ball and 0 otherwise. sM=(0,-1,0)
After 10 iterations, the final values are between 0 and 2.88 reached at the point +M of the graphic
The energy value is 4.23 . Epsilon=10-3
The ground state is concentrated mainly on the left ball.


The initial value u0(x) is 0 on the balls and 1 otherwise. sM=(0,0.0243,0)
After 100 iterations (non convergence), the final values are between 0 and 4.75 reached at the point +M of the graphic
The energy value is 5.21 . Epsilon=10-3
The ground state is concentrated mainly on the left ball.


The initial value u0(x) is 1 on the large ball and 0 otherwise. sM=(0,2,0)
After 21 iterations, the final values are between 0 and 2.38 reached at the point +M of the graphic
The energy value is 6.57 . Epsilon=10-3
This solution is concentrated mainly into the large ball under and at right of the small ball hole.

Different values of the exponent p:
What happens to numerical solutions when the exponent p of up, (in fact, |u|p-1 u ) has a value greater or equal of the critical exponent p* = (d+2)/(d-2) = (3+2)/(3-2) = 5 in 3d for a star-shaped domain?

Here, the domain is not in this condition with the presence of a cavity and balls.
The works of Bahri-Coron and Ding indicate that in this case, the equation may have multiple solutions.
Numerically, this is verified with different exponents p>=5.

The mesh has 6257 vertices, 44640 nodes and 29554 tetrahedra.
The maximum of the initial solution is 1.00001 at (0, 2, 0).
The initial solution is equal to zero outside the great sphere and 1 inside.
The homogeneous Dirichlet boundary condition is used.
The cut off convergence error indicator is 10-3
The cpu time is about 10 seconds.
The preconditioned conjuguate gradient requires 8 706 832 words in main memory.
Exponent p=4.5
27 iterations have been necessary.
The energy=7.136 and the solution maximum 7.108 is reached at ( 0, 1.934, -1.046 ).


Exponent p=4.9
131 iterations with oscillations have been necessary.
The energy=5.165 and the solution maximum 9.596 is reached at ( 0, 1.934, -0.0697 ).



Exponent p=5
92 iterations with oscillations have been necessary.
The energy=4.766 and the solution maximum 9.934 is reached at ( 0, 1.934, -0.0698 ).



Exponent p=5.1
73 iterations with oscillations have been necessary.
The energy=4.400 and the solution maximum 10.019 is reached at ( 0, 1.934, -0.0698 ).



Exponent p=5.5
44 iterations have been necessary.
The energy=3.245 and the solution maximum 9.327 is reached at ( 0, 1.934, -0.0698 ).


Exponent p=8.0
19 iterations have been necessary.
The energy=0.9192 and the solution maximum 4.872 is reached at ( 0, 1.909, -0.0698 ).



Exponent p=50
10 iterations with oscillations have been necessary.
The energy=0.0907 and the solution maximum 1.419 is reached at ( 0, 1.786, -0.0349 ).


Conclusions:

More details are given in the publication:
G. Chen, W.M. Ni, A. Perronnet, J. Zhou: Algorithms and Visualization for solutions of Nonlinear Elliptic Equations
Part II: Dirichlet, Neumann and Robin Boundary Conditions and Problems in 3d
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Vol. 11, No 7 (2001), pages 1781-1799

The Mefisto-MAILLER input data file of test/lane3d

The Mefisto-THERMICER input data file of test/lane3d
Page written by Alain Perronnet. Last update April 28-th 2009