Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation
Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation
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DOI：
10.1007/s00211-022-01318-8
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发表时间：
2021-06
影响因子：
2.1
通讯作者：
Shihua Gong;M. Gander;I. Graham;D. Lafontaine;E. Spence
中科院分区：
文献类型：
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作者：
Shihua Gong;M. Gander;I. Graham;D. Lafontaine;E. Spence
我们分析了亥姆霍兹方程的并行重叠施瓦茨域分解方法，其中子域之间的信息交换是使用一阶吸收（阻抗）传输条件以及单位划分来实现的。我们在偏微分方程（PDE）层面上对该方法进行了新颖的分析（无离散化）。首先，我们将该方法表述为定点迭代，并表明（在维度 1、2、3 中）该方法在适当局部函数空间的张量积中得到了明确定义，每个函数空间都有阻抗边界数据。然后，我们根据子域之间局部相互作用产生的某些阻抗到阻抗图的局部范数，获得定点算子范数的界限。这些界限提供了定点算子（的某些幂）是收缩的条件。在二维中，对于矩形域和条状域分解（每个子域仅与其直接相邻的子域重叠），我们提出了两种技术来验证阻抗到阻抗图的假设，以确保定点算子的幂收缩性。第一种是通过半经典分析，当频率趋于无穷大时，它会给出有效的严格估计。至少对于具有两个子域的模型情况，这些结果验证了足够大重叠所需的假设。对于更现实的域分解，我们通过解决某些规范（局部）特征值问题来直接计算阻抗到阻抗图的范数。我们给出了数值实验来说明该理论。这些还表明，迭代方法保持收敛和/或在理论未涵盖的情况下提供良好的预处理器，包括一般域分解，例如通过自动图分区软件获得的分解。
We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1, 2, 3) that it is well-defined in a tensor product of appropriate local function spaces, each withimpedance boundary data. We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance maps that ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. At least for a model case with two subdomains, these results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedance-to-impedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graph-partitioning software.