标题： 一种二维任意形状泊松方程的新型高效数值解法 显示英文标题

标题： A novel efficient numerical solution of Poisson equation for arbitrary shapes in two dimensions

摘要： 我们提出了一种新颖的高效算法，用于求解静电学中的不规则二维区域的泊松方程。 它可以处理狄利克雷、诺伊曼或混合边界问题，在这些问题中填充介质可以是均匀的或非均匀的。 新方法的基本思想是分三步求解：(i) 首先求解方程 $\nabla\cdot\mathbf D=\rho$。 在限制子空间中，散度算子的逆被找到，通过快速直接求解器在 O(N) 操作中得到电通量密度 $\mathbf D$。 所得到的 $\mathbf D$是非唯一的，并且具有不确定的无散成分。 然后通过 $\mathbf E=\mathbf D/\epsilon$得到电场。 但是 $\nabla\times\mathbf E=0$用于静电场；因此， $\mathbf E$是无旋的，并且与无散空间正交。 (ii) 使用正交化过程来净化电场，使其无旋且唯一。 (iii) 然后通过求解$\phi$或在受限子空间中通过类似的快速直接求解器找到梯度算子的逆，从而得到电势$\nabla \phi=-\mathbf E$。处理了狄利克雷和诺伊曼边界条件的情况。最后，通过几个数值例子说明了验证性和效率。通过这些模拟，观察到所提出方法的计算复杂度几乎按 O(N) 缩放，其中 N 是网格的三角形片数。因此，这种新算法是一种可行的快速泊松求解器。 显示英文摘要

摘要： We propose a novel efficient algorithm to solve Poisson equation in irregular two dimensional domains for electrostatics. It can handle Dirichlet, Neumann or mixed boundary problems in which the filling media can be homogeneous or inhomogeneous. The basic idea of the new method is solve the problem in three steps: (i) First solve the equation $\nabla\cdot\mathbf D=\rho$. The inverse of the divergence operator in a restricted subspace is found to yield the electric flux density $\mathbf D$ by a fast direct solver in O(N) operations. The $\mathbf D$ so obtained is nonunique with indeterminate divergence-free component. Then the electric field is found by $\mathbf E=\mathbf D/\epsilon$. But $\nabla\times\mathbf E=0$ for electrostatic field; hence, $\mathbf E$ is curl free and orthogonal to the divergence free space. (ii) An orthogonalization process is used to purify the electric field making it curl free and unique. (iii) Then the potential $\phi$ is obtained by solving $\nabla \phi=-\mathbf E$ or finding the inverse of the gradient operator in a restricted subspace by a similar fast direct solver in O(N) operations. Treatments for both Dirichlet and Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, it is observed that the computational complexity of our proposed method almost scales as O(N) where N is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.