An Efficient Unconditionally Stable Method for Dirichlet Partitions in Arbitrary Domains

A Dirichlet $k$-partition of a domain is collection $k$ pairwise disjoint open subsets such that the sum their first Laplace--Dirichlet eigenvalues minimal. In this paper, we propose new relaxation problem by introducing auxiliary indicator functions domains and develop simple efficient diffusion generated method to compute $k$-partitions for arbitrary domains. The only alternates three steps: (1) convolution, (2) thresholding, (3) projection. simple, easy implement, insensitive initial guesses can be effectively applied without any special discretization. At each iteration, computational complexity linear in discretization domain. Moreover, theoretically prove energy decaying property method. Experiments are performed show accuracy approximation, efficiency, unconditional stability algorithm. We apply proposed algorithms on both 2- 3-dimensional flat tori, triangle, square, pentagon, hexagon, disk, three-fold star, five-fold cube, ball, tetrahedron different effectiveness Compared previous work with reported time, achieves hundreds times acceleration.