The Dirichlet–Neumann alternating method (D-N method) is considered for solving the Poisson equation in a two dimensional unbounded domain. The method is a non-overlapping domain decomposition method for alternately solving two boundary value problems in two decomposed subdomains by introducing an artificial boundary. The iterative solution obtained by the method converges to the exact one for a suitably chosen relaxation parameter, which determines the speed of convergence. The purpose of this paper is to derive some mathematical formulas for the relaxation parameter to further improve the speed of convergence compared with the existing optimal one. The method of fundamental solutions with unconventional basis functions satisfying a condition at infinity is applied for solving the Laplace equation in the unbounded subdomain of no interest for effectively using the D-N method. Since the artificial boundary is taken as a circle, the coefficient matrix of the resultant system of linear equations is a circulant matrix. The fast Fourier transform can then be applied to this system of linear equations to obtain the solution without directly solving it. Numerical experiments show that the improved relaxation parameters can yield faster convergence than the conventional optimal one.

中文翻译：

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非重叠域分解方法的数学改进收敛分析


狄利克雷-诺依曼交替法（DN 法）被考虑用于求解二维无界域中的泊松方程。该方法是一种非重叠域分解方法，通过引入人工边界交替求解两个分解子域中的两个边值问题。对于适当选择的松弛参数，该方法获得的迭代解收敛到精确解，松弛参数决定收敛速度。本文的目的是推导一些松弛参数的数学公式，以比现有的最优收敛速度进一步提高。采用非常规基函数满足无穷远条件的基本解法来求解无界子域中的拉普拉斯方程，以有效利用DN方法。由于人工边界取为圆，因此所得线性方程组的系数矩阵是循环矩阵。然后可以将快速傅立叶变换应用于该线性方程组以获得解，而无需直接求解。数值实验表明，改进的松弛参数比传统的最优松弛参数能够产生更快的收敛速度。