Abstract: The procedure of multigrid method is described in detail and its properties are discussed by applying multigrid to both positive definite Helmholtz equation and indefinite Helmholtz equation. Comparisons of convergence performance are implemented between V cycle, W cycle and F cycle. It is shown that, by solving positive definite Helmholtz equation, multigrid method has high computing efficiency. However, as the value of wave-number increases in indefinite Helmholtz equation, the results obtained by multigrid method diverge. The reasons lie in both fine grid smoothing and coarse grid correction, and the improvement of Multigrid is still needed in future.

Key words: multigrid, F cycle, positive definite Helmholtz, indefinite Helmholtz

摘要： 详细给出了多重网格方法的实现过程，借助正定Helmholtz方程及不定Helmholtz方程的求解来探讨多重网格方法的特性。对多重网格V环、W环以及F环三种不同迭代格式的收敛效果进行了对比。通过正定Helmholtz方程的求解，发现多重网格的确有很高的计算效率。对于不定Helmholtz方程，随着波数的增加，利用多重网格方法得到结果不收敛，原因出在细网格光滑和粗网格矫正过程。如何针对此问题对多重网格进行有效改进还有待进一步研究。

关键词: 多重网格, F环, 正定Holmholtz方程, 不定Helmholtz方程