Schur complement Domain Decomposition Methods for the solution of multiple scattering problems
Schur complement Domain Decomposition Methods for the solution of multiple scattering problems
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解决多重散射问题的 Schur 补域分解方法
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Y. Boubendir
中科院分区：
文献类型：
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作者：
Michel Pedneault;C. Turc;Y. Boubendir
提出了一种求解频域多次散射问题的Schur补域分解算法。就像在经典的DD方法中一样，我们(1)将散射体集合封闭在一个以人工边界为边界的区域中，(2)我们将该区域细分为一个不重叠的子域集合，使得子域的边界不与任何散射体相交，以及(3)我们通过在子域之间的公共界面上匹配Robin边界条件来连接子问题的解。我们使用子域Robin-to-Robin映射将DD问题重塑为一个稀疏线性系统，其未知由子域之间接口上的Robin数据组成-每个接口两个未知数。Robin-to-Robin映射是用条件良好的边界积分算子来计算的。与经典DD不同的是，我们不以不动点迭代的形式重新表述区域分解问题，而是通过Schur补对子域之间的内部界面对应的未知数进行高斯消去来求解随后的线性系统。一旦所有对应于内部子域界面的未知数都被消除，我们就求解了一个涉及内外人工边界上的未知数的小得多的线性系统。我们给出了数值证据，证明我们的Schur补码DD算法可以产生其他现有方法所无法获得的非常大的多次散射问题的精确解。
We present a Schur complement Domain Decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods we (1) enclose the ensemble of scatterers in a domain bounded by an artificial boundary, (2) we subdivide this domain into a collection of nonoverlapping subdomains so that the boundaries of the subdomains do not intersect any of the scatterers, and (3) we connect the solutions of the subproblems via Robin boundary conditions matching on the common interfaces between subdomains. We use subdomain Robin-to-Robin maps to recast the DD problem as a sparse linear system whose unknown consists of Robin data on the interfaces between subdomains---two unknowns per interface. The Robin-to-Robin maps are computed in terms of well-conditioned boundary integral operators. Unlike classical DD, we do not reformulate the Domain Decomposition problem in the form a fixed point iteration, but rather we solve the ensuing linear system by Gaussian elimination of the unknowns corresponding to inner interfaces between subdomains via Schur complements. Once all the unknowns corresponding to inner subdomains interfaces have been eliminated, we solve a much smaller linear system involving unknowns on the inner and outer artificial boundary. We present numerical evidence that our Schur complement DD algorithm can produce accurate solutions of very large multiple scattering problems that are out of reach for other existing approaches.