Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
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DOI：
10.1016/j.cam.2018.11.035
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发表时间：
2018-06
期刊：
影响因子：
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通讯作者：
G. Diwan;A. Moiola;E. Spence
中科院分区：
文献类型：
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作者：
G. Diwan;A. Moiola;E. Spence
Moiola和Spence（2014）引入了亥姆霍兹方程的一个新的强制公式。在本文中，我们研究了这个公式的h版伽辽金离散，以及由此产生的线性系统的迭代解。我们发现强制公式在污染效应方面的表现与标准公式相似（即为了保持k→∞时的精度，h必须以与标准公式相同的速率随k减小）。我们证明了在给定对称正定矩阵的前提下求解新公式的线性系统所需的GMRES迭代次数的k显式界。尽管迭代次数随着k的增长而增长，但对于亥姆霍兹方程的预条件公式来说，这是第一个如此严格的GMRES迭代次数界限，其中预条件是对称正定矩阵。
A new, coercive formulation of the Helmholtz equation was introduced in Moiola and Spence (2014). In this paper we investigate h-version Galerkin discretisations of this formulation, and the iterative solution of the resulting linear systems. We find that the coercive formulation behaves similarly to the standard formulation in terms of the pollution effect (ie to maintain accuracy as k→∞, h must decrease with k at the same rate as for the standard formulation). We prove k-explicit bounds on the number of GMRES iterations required to solve the linear system of the new formulation when it is preconditioned with a prescribed symmetric positive-definite matrix. Even though the number of iterations grows with k, these are the first such rigorous bounds on the number of GMRES iterations for a preconditioned formulation of the Helmholtz equation, where the preconditioner is a symmetric positive-definite matrix.