Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?
Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?
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DOI：
10.1007/s00211-015-0700-2
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发表时间：
2015-11-01
影响因子：
2.1
通讯作者：
Spence, E. A.
中科院分区：
文献类型：
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作者：
Gander, M. J.;Graham, I. G.;Spence, E. A.
最近有很多关于亥姆霍兹算子的预处理离散化的研究（适用于合适的边界条件），使用所谓的“移位拉普拉斯算子”的离散版本。这是由以下事实驱动的：随着增加，转移的问题变得更容易迭代求解。尽管有许多数值研究，但没有严格的分析如何选择移位。在本文中，我们专注于如何大的问题，使移位问题提供了一个预条件，导致GMRES的独立收敛，我们的主要结果是一个充分条件，这个属性举行。这一结果适用于有限元离散的内部阻抗问题和软声散射问题（与辐射条件在后一个问题作为一个远场阻抗边界条件）。注意，我们并没有解决这样一个重要的问题，即预条件子应该有多大，才能很容易地被标准迭代方法求逆。
There has been much recent research on preconditioning discretisations of the Helmholtz operator (subject to suitable boundary conditions) using a discrete version of the so-called "shifted Laplacian" for some . This is motivated by the fact that, as increases, the shifted problem becomes easier to solve iteratively. Despite many numerical investigations, there has been no rigorous analysis of how to chose the shift. In this paper, we focus on the question of how large can be so that the shifted problem provides a preconditioner that leads to -independent convergence of GMRES, and our main result is a sufficient condition on for this property to hold. This result holds for finite element discretisations of both the interior impedance problem and the sound-soft scattering problem (with the radiation condition in the latter problem imposed as a far-field impedance boundary condition). Note that we do not address the important question of how large should be so that the preconditioner can easily be inverted by standard iterative methods.