On a Class of Preconditioners for Solving the Helmholtz Equation
On a Class of Preconditioners for Solving the Helmholtz Equation
复制标题
DOI：
10.1016/j.apnum.2004.01.009
复制
发表时间：
2003-06
期刊：
影响因子：
--
通讯作者：
Y. Erlangga;C. Vuik;C. Oosterlee
中科院分区：
文献类型：
--
作者：
Y. Erlangga;C. Vuik;C. Oosterlee
在1983年，提出了一个预条件[J. COMPUT。Phys. 49（1983）443]基于拉普拉斯算子，用CGNR有效地求解离散亥姆霍兹方程。预条件是特别有效的低波数的情况下，线性系统是轻微的不确定性。Laird [Preconditioned iterative solution of the 2D Helmholtz equation，First Year's Report，St. Hugh's College，Oxford，2001]提出了一种预处理器，其中将额外的项添加到拉普拉斯算子。这一项类似于亥姆霍兹方程中的零阶项，但符号相反。本文将这两种方法进一步推广到一类新的预条件子，即所谓的“移位拉普拉斯”预条件子，其形式为Δφ−αk2φ，其中α∈ C。对不同波数的数值实验表明了预处理器的有效性。预条件子与GMRES、Bi-CGSTAB和CGNR结合进行评估。
In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called “shifted Laplace” preconditioners of the form Δφ−αk2φ with α∈ C . Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.