In this paper, we study a rearrangement method for solving a maximization problem associated with Poisson’s equation with Dirichlet boundary conditions. The maximization problem is to find the forcing within a certain admissible set as to maximize the total displacement. The rearrangement method alternatively (i) solves the Poisson equation for a given forcing and (ii) defines a new forcing corresponding to a particular super-level-set of the solution. Rearrangement methods are frequently used for this problem and a wide variety of similar optimization problems due to their convergence guarantees and observed efficiency; however, the convergence rate for rearrangement methods has not generally been established. In this paper, for the one-dimensional problem, we establish linear convergence. We also discuss the higher dimensional problem and provide computational evidence for linear convergence of the rearrangement method in two dimensions.

在本文中，我们研究了一种重排方法，用于解决与Dirichlet边界条件下的泊松方程有关的最大化问题。最大化问题是在一定的允许范围内找到强迫，以使总位移最大化。重新排列方法可以选择（i）求解给定强迫的泊松方程，并且（ii）定义对应于解决方案的特定超水平集的新强迫。由于它们的收敛性保证和观察到的效率，重排方法经常用于这个问题和各种各样的类似优化问题。但是，重排方法的收敛速度尚未确定。在本文中，对于一维问题，我们建立了线性收敛。