A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator
A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator
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DOI：
10.1016/j.cam.2014.03.031
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发表时间：
2014-12
期刊：
影响因子：
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通讯作者：
L. Conen;V. Dolean;R. Krause;F. Nataf
中科院分区：
文献类型：
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作者：
L. Conen;V. Dolean;R. Krause;F. Nataf
控制波传播和散射现象的亥姆霍兹方程很难用数值求解。其采用分段线性有限元的离散化会产生典型的大型线性方程组。因此，本质上并行的域分解方法构成了一类有前途的预处理器。这些方法的一个基本要素是良好的粗糙空间。在这里，亥姆霍兹方程提出了一个特殊的挑战，因为即使与最优选择的轻微偏差也可能是毁灭性的。在本文中，我们提出了一个基于涉及狄利克雷到诺依曼算子的局部特征问题的粗糙空间。我们的构建是完全自动的，无需调整参数即可确保良好的收敛速度。此外，它自然地考虑波数的局部变化，因此也适用于异构亥姆霍兹问题。由此产生的方法在设计上是并行的，并且其效率在二维同质和异质数值示例中得到了证明。
The Helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically. Its discretization with piecewise linear finite elements results in typically large linear systems of equations. The inherently parallel domain decomposition methods constitute hence a promising class of preconditioners. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be devastating.In this paper, we present a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples.