When considered as a standalone iterative solver for elliptic boundary value problems, the Dirichlet-Neumann (DN) method is known to converge geometrically for domain decompositions into strips, even for a large number of subdomains. However, whenever the domain decomposition includes cross-points, i.e.$\!$ points where more than two subdomains meet, the convergence proof does not hold anymore as the method generates subproblems that might not be well-posed. Focusing on a simple two-dimensional example involving one cross-point, we proposed in a previous work a decomposition of the solution into two parts: an even symmetric part and an odd symmetric part. Based on this decomposition, we proved that the DN method was geometrically convergent for the even symmetric part and that it was not well-posed for the odd symmetric part. Here, we introduce a new variant of the DN method which generates subproblems that remain well-posed for the odd symmetric part as well. Taking advantage of the symmetry properties of the domain decomposition considered, we manage to prove that our new method converges geometrically in the presence of cross-points. We also extend our results to the three-dimensional case, and present numerical experiments that illustrate our theoretical findings.

翻译：当将Dirichlet-Neumann(DN)方法视为椭圆边值问题的独立迭代求解器时，已知对于带状区域分解，即使子域数量众多，该方法也能几何收敛。然而，当区域分解包含交叉点（即多于两个子域相交的点）时，由于该方法可能生成不适定的子问题，收敛证明不再成立。针对一个包含单一交叉点的简单二维示例，我们在先前工作中提出将解分解为两部分：偶对称部分和奇对称部分。基于此分解，我们证明了DN方法对偶对称部分几何收敛，而对奇对称部分不适定。本文引入DN方法的一种新变体，该变体对于奇对称部分也能生成保持适定的子问题。利用所考虑区域分解的对称性质，我们成功证明了新方法在存在交叉点时几何收敛。我们还将结果推广到三维情形，并给出数值实验以佐证理论发现。