Mathematics > Numerical Analysis
[Submitted on 26 Aug 2021 (v1), last revised 29 Mar 2022 (this version, v2)]
Title:An efficient unconditionally stable method for Dirichlet partitions in arbitrary domains
View PDFAbstract:A Dirichlet $k$-partition of a domain is a collection of $k$ pairwise disjoint open subsets such that the sum of their first Laplace--Dirichlet eigenvalues is minimal. In this paper, we propose a new relaxation of the problem by introducing auxiliary indicator functions of domains and develop a simple and efficient diffusion generated method to compute Dirichlet $k$-partitions for arbitrary domains. The method only alternates three steps: 1. convolution, 2. thresholding, and 3. projection. The method is simple, easy to implement, insensitive to initial guesses and can be effectively applied to arbitrary domains without any special discretization. At each iteration, the computational complexity is linear in the discretization of the computational domain. Moreover, we theoretically prove the energy decaying property of the method. Experiments are performed to show the accuracy of approximation, efficiency and unconditional stability of the algorithm. We apply the proposed algorithms on both 2- and 3-dimensional flat tori, triangle, square, pentagon, hexagon, disk, three-fold star, five-fold star, cube, ball, and tetrahedron domains to compute Dirichlet $k$-partitions for different $k$ to show the effectiveness of the proposed method. Compared to previous work with reported computational time, the proposed method achieves hundreds of times acceleration.
Submission history
From: Dong Wang [view email][v1] Thu, 26 Aug 2021 02:25:14 UTC (8,309 KB)
[v2] Tue, 29 Mar 2022 00:30:56 UTC (17,731 KB)
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