Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems
Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems
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DOI：
10.1090/s0025-5718-06-01930-2
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发表时间：
2006-12
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影响因子：
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通讯作者：
J. Bramble;J. Pasciak
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文献类型：
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作者：
J. Bramble;J. Pasciak
本文研究了用截断域完全匹配层（PML）近似频域三维麦克斯韦散射问题。我们还处理的时间谐波PML近似的声散射问题。继1998年Lassas和Somersalo的工作之后，基于球面几何定义了一个过渡层，这导致了过渡层外的常系数问题。然后定义一个截断（计算）域，它覆盖了过渡区域。截断域仅需要具有最小平滑的外边界（例如，Lipschitz连续）。我们考虑截断的PML问题的结果时，一个完美的导电边界条件施加在截断域的外边界。截断PML问题的解的存在性和唯一性将被证明，只要截断域足够大，例如，包含一个半径为Rt的球体。我们还显示指数（在参数R t）收敛截断PML解决方案的原始散射问题的解决方案内的过渡层。我们的结果是重要的，因为他们是第一个表明，截断PML问题可以提出一个区域上的非光滑外边界。这允许使用基于多边形网格的近似。此外，即使过渡系数取决于球面几何形状，他们可以任意光滑，因此产生的问题是服从数值求积。基于我们的分析的近似方案是未来研究的重点。
We consider the approximation of the frequency domain three-dimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius R t . We also show exponential (in the parameter R t ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer. Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.