论文标题
在优化的施瓦茨方法中对跨点的强大处理
Robust treatment of cross points in Optimized Schwarz Methods
论文作者
论文摘要
在域分解（DD）领域中，优化的Schwarz方法（OSM）似乎是解决大型时谐波传播问题的重要技术之一。它基于适当的传输条件，使用精心设计的阻抗操作员在子域之间交换信息。然而，这种方法的效率受到跨点的存在的阻碍，如果没有适当的治疗，则有两个以上的子域基座。在这项工作中，我们提出了Helmholtz方程的交叉点问题的新处理方法，该问题在任何几何界面配置中均保持有效。我们利用多条形形式主义来定义具有合适的连续性和等轴测特性的新交换运算符。然后，我们开发一个完整的理论框架，将经典OSM推广到具有跨点的分区，并包含严格的几何融合证明，相对于网格离散化统一，以提供适当的积极阻抗操作员。提供了2D和3D中的广泛数值结果，以说明所提出方法的性能。
In the field of Domain Decomposition (DD), Optimized Schwarz Method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between sub-domains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two sub-domains abut, if no appropriate treatment is provided. In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the performance of the proposed method.
