论文标题
基于痕量转移的对角线扫描域分解方法用于Helmholtz方程：算法和收敛分析
Trace Transfer-based Diagonal Sweeping Domain Decomposition Method for the Helmholtz Equation: Algorithms and Convergence Analysis
论文作者
论文摘要
通过利用完美匹配的层（PML）和源传输技术，最近开发了对角线扫描域分解方法（DDM），用于求解$ \ Mathbb {r}^n $中的高频Helmholtz方程，该方程使用$ 2^n $ scheeps in diagonal Diagonal Docompositions conferpositions diagonal domomain dompositions pockositions conforpositions pockositions conforpositions。尽管这种对角线扫描DDM本质上是乘法的，但与管道处理相结合时，它非常适合与多个右侧的Helmholtz问题并行计算，因为每个扫描中的顺序步骤的数量远小于子域的数量。在本文中，我们提出和分析了基于痕量转移的对角线清除DDM。从源传输转换为邻居子域之间的信息传递的主要优点是，由于转移的轨迹在邻居子域之间仅具有$ 2N $的主要方向，而转移的sivers则来自转移的底部，而从总计$ 3^n-1 $ cardinal和Corner方向则更容易分析和实施，因此更容易分析和实施。我们严格地证明，所提出的对角扫描DDM不仅在恒定的中型情况下给出了全局PML问题的精确解决方案，而且在两层介质案例中最多可以将其与最多一轮的对角线扫描一起使用，这是该方法的理论基础。还通过在两个和三个维度的大量实验中证明了所提出的DDM作为直接求解器或预处理程序的性能和并行可伸缩性。
By utilizing the perfectly matched layer (PML) and source transfer techniques, the diagonal sweeping domain decomposition method (DDM) was recently developed for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$, which uses $2^n$ sweeps along respective diagonal directions with checkerboard domain decomposition. Although this diagonal sweeping DDM is essentially multiplicative, it is highly suitable for parallel computing of the Helmholtz problem with multiple right-hand sides when combined with the pipeline processing since the number of sequential steps in each sweep is much smaller than the number of subdomains. In this paper, we propose and analyze a trace transfer-based diagonal sweeping DDM. A major advantage of changing from source transfer to trace transfer for information passing between neighbor subdomains is that the resulting diagonal sweeps become easier to analyze and implement and more efficient, since the transferred traces have only $2n$ cardinal directions between neighbor subdomains while the transferred sources come from a total of $3^n-1$ cardinal and corner directions. We rigorously prove that the proposed diagonal sweeping DDM not only gives the exact solution of the global PML problem in the constant medium case but also does it with at most one extra round of diagonal sweeps in the two-layered media case, which lays down the theoretical foundation of the method. Performance and parallel scalability of the proposed DDM as direct solver or preconditioner are also numerically demonstrated through extensive experiments in two and three dimensions.
