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).