Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian
Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian
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DOI：
10.1137/060661491
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发表时间：
2007-09
期刊：
影响因子：
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通讯作者：
M. Gijzen;Y. Erlangga;C. Vuik
中科院分区：
文献类型：
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作者：
M. Gijzen;Y. Erlangga;C. Vuik
移位拉普拉斯预条件子作为一种加速Helmholtz方程迭代解法收敛的技术，引起了人们的广泛关注。在本文中，我们提出了一个全面的谱分析的Helmholtz算子预处理与移位拉普拉斯算子。我们的分析在一般情况下是有效的。传播介质可以是异质的，并且分析也适用于不同类型的阻尼，包括计算域边界的辐射条件。通过将预处理Helmholtz算子的谱分析结果与GMRES-剩余范数的上界相结合，我们能够提供移位的最佳值，并解释了用移位拉普拉斯算子预处理的GMRES收敛的网格依赖性。我们说明我们的结果与地震测试问题。
Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm we are able to provide an optimal value for the shift, and to explain the meshdependency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem.