This paper explores a family of generalized sweeping preconditionners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (e.g. horizontal and diagonal). Several directions can be combined thanks to the flexible version of GMRES, allowing for the rapid transfer of information in the different zones of the computational domain, then accelerating the convergence of the final iterative solution procedure. Several two-dimensional finite element results are proposed to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity.
翻译:本文探索了一套关于赫姆霍茨在计算域中非重叠检查板分割方面存在普遍而全面问题的先决条件; 域分解程序依赖于高顺序传输条件和交叉点处理,如果没有在子域数量增加时有效的先决条件技术,这些条件和跨点处理就无法扩大规模; 采用拟议办法,现有的对称高斯-赛德尔和平行双扫前题等全面先决条件可以适用于具有不同横幅方向(例如横向和对角)的检查板隔板; 由于GMRES的灵活版本,可以将若干方向结合起来,从而允许在计算域的不同区域迅速传递信息,然后加快最后迭代解答程序的趋同; 提议采用若干两维的有限要素,以研究和比较全面先决条件,并展示在日益复杂的情况下的表现。