Candidates for the 3-minimal partition
Algorithm to propose a candidate for the minimal 3-partition:
Initialisation.
Let
D
0
=(D
1
0
,D
2
0
,D
3
0
) be a 3-partition of Ω
Iteration.
For n≥1, we define the partition
D
n+1
=(Ω
1
n
,Ω
2
n
,Ω
3
n
) by
Ω
1
n
= Ω
3
n-1
(Ω
2
n
,Ω
3
n
) is the nodal partition associated to the second eigenvector of the Dirichlet-Laplacian on Int(Ω∖Ω
1
n
)
We denote λ
j
(Ω
k
) the j-th eigenvalue of the Dirichlet-Laplacian on Ω
k
and for n≥0, Λ(
D
n
) = max(λ
1
(Ω
1
n
),λ
1
(Ω
2
n
),λ
1
(Ω
3
n
))
Remark.
If the algorithm converges to the partition
D
=(Ω
1
,Ω
2
,Ω
3
), then Λ(
D
) = λ(Ω
1
) = λ(Ω
2
) = λ(Ω
3
)
Reference.
Farid Bozorgnia,
Numerical algorithms for the spatial segregation of competitive systems
, submitted for publication (2008).
Disk of area 1 with coarse mesh
Disk of area 1 with fine mesh
Square of area 1 with coarse mesh
Square of area 1 with fine mesh
Equilateral triangle of area 1 with coarse mesh
Equilateral triangle of area 1 with fine mesh
3-hexagons of area 1 with coarse mesh
3-hexagons of area 1 with fine mesh