Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media
Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media
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DOI：
10.1007/s10915-020-01133-z
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发表时间：
2018-09
影响因子：
2.5
通讯作者：
D. Nicholls;Carlos P'erez-Arancibia;C. Turc
中科院分区：
文献类型：
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作者：
D. Nicholls;Carlos P'erez-Arancibia;C. Turc
我们提出了一种适用于周期性分层介质中亥姆霍兹传输问题的准最优域分解方法（DD）的全面预处理器。亥姆霍兹方程的准最优 DD (QO DD) 依赖于狄利克雷到诺伊曼 (DtN) 算子的近似传输算子。利用形状扰动级数，我们构造了与周期域相对应的 DtN 算子的近似值，然后将其用作非重叠 DD 框架中的传输算子。作为 DD 构建块的 Robin-to-Robin (RtR) 算子通过稳健的边界积分方程公式来表达。我们使用准周期边界积分算子的 Nyström 离散化来构造 RtR 的高阶近似。基于准最优传输算子应该像完美透明边界条件一样的前提，我们构造了与周期性分层介质相关的三对角 QO Schwarz 迭代矩阵的近似 LU 分解，然后将其用作双扫描预处理器。我们提出了各种数值结果，展示了应用于 QO DD 的扫描预处理器对于周期性层状介质中亥姆霍兹传输问题的迭代解决方案的有效性。
We present a sweeping preconditioner for quasi-optimal domain decomposition methods (DD) applied to Helmholtz transmission problems in periodic layered media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission operators that are approximations of Dirichlet-to-Neumann (DtN) operators. Employing shape perturbation series, we construct approximations of DtN operators corresponding to periodic domains, which we then use as transmission operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators that are the building blocks of DD are expressed via robust boundary integral equation formulations. We use Nyström discretizations of quasiperiodic boundary integral operators to construct high-order approximations of RtR. Based on the premise that the quasi-optimal transmission operators should act like perfect transparent boundary conditions, we construct an approximate LU factorization of the tridiagonal QO Schwarz iteration matrix associated with periodic layered media, which is then used as a double sweep preconditioner. We present a variety of numerical results that showcase the effectiveness of the sweeping preconditioners applied to QO DD for the iterative solution of Helmholtz transmission problems in periodic layered media.