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.

翻译：域的 Dirichlet $k 分割区是一个域的配方, 配方是 $k$ 的集合, 配方是 $k$ 的配方, 配方是 折叠, 2 阈值, 和 3. 投影 。 方法简单易行, 容易执行, 不理会初始猜想, 可以有效地应用于任意的域, 而没有特殊的离散 。 在每一个迭代中, 计算复杂程度在计算域的离散中是线性的 。 此外, 我们理论上证明该方法的能量衰变属性。 进行实验以显示算法的近似性、 效率和无条件稳定性 。 我们对二维和三维平坦、 三角、 方、 方形、 方形、 六边、 盘、 三维恒星、 五维星、 立方、 球、 四重的计算复杂度, 以上方计算方法显示前一个数百美元的加速度计算法 。