Minimal perimeter equal-area partitions of 2D regions

In order to numerically approximate equal-area partitions of 2D regions which minimize the perimeter we use a $\Gamma$-convergence approximation. (See this link for an introduction to the use of Gamma-convergence methods in numerical computations. Related topics are: link1, link2

I approximate the sum of the perimeter with a sum of corresponding Modica-Mortola functionals. The proof of the $\Gamma$-convergence result is not straightforward and not trivial, since the $\Gamma$-convergence is not stable for the sum. The numerical approach consists of triangulating the domain, computing the rigidity and mass matrices associated to the P1 finite elements, and then use these matrices to evaluate every piece of the approximating functional.

I use the method proposed by E. Oudet (link) to non-rectangular domains by using a finite elements approach. This approach gives better results. In particular, I obtain similar configuration as the ones obtained by Cox and Fikkema (link).

If instead considering equal areas, we impose different fixed area conditions, we can study, for example, problems concerning bubble clusters.

Created: Dec 2014