An algebraic multigrid (AMG) with aggregation technique to coarsen is applied to construct a better preconditioner for solving Helmholtz equations in this paper. The solution process consists of constructing the preconditioner by AMG and solving the preconditioned Helmholtz problems by Krylov subspace methods. In the setup process of AMG, we employ the double pairwise aggregation (DPA) scheme firstly proposed by Y. Notay (2006) as the coarsening method. We compare it with the smoothed aggregation algebraic multigrid and meanwhile show shifted Laplacian preconditioners. According to numerical results, we find that DPA algorithm is a good choice in AMG for Helmholtz equations in reducing time and memory. Spectral estimation of system preconditioned by the three methods and the influence of second-order and fourth-order accurate discretizations on the three techniques are also considered.
Dandan Chen. Ting-Zhu Huang. Liang Li. "Comparison of Algebraic Multigrid Preconditioners for Solving Helmholtz Equations." J. Appl. Math. 2012 (SI12) 1 - 12, 2012. https://doi.org/10.1155/2012/367909