求解Helmholtz方程的新型稀疏近似逆预条件算法

A New Approximate Inverse Preconditioning Algorithm for Solving Helmholtz Equation

提出了一种新型预条件算法,用于对有限元法离散Helmholtz方程所产生的大型稀疏复对称且高度不定的线性系统进行高效迭代求解。该新型预条件子是在复拉普拉斯偏移算子的基础上结合改进的稀疏近似逆算法来得到。通过改善矢量有限元线性系统自身的谱特性,该预条件算法既可避免迭代中的不稳定情况,同时也能较大提高迭代求解效率。数值结果表明,与若干常用预条件算法相比,所提出的预条件算法更加有效。

A new preconditioning algorithm is presented for effectively solving the large complex symmetric and always highly indefinite linear equations arising from the finite element method（FEM） discretizing Helmholtz equation.The proposed preconditioner is constructed by combing the complex shifted Laplace（CSL） operator with a modified AINV（MAINV） approximate inverse algorithm.By improving the FEM linear system＇s eigenvalue spectrum,the proposed preconditioner can avoid most of the breakdowns during the iterative process,and enhance the solving efficiency.Numerical examples demonstrate the proposed preconditioning algorithm is more effective than some standard ones.