To Solve The Helmholtz Equation Of Algebraic Multigrid Preconditioned Technology Research

Due to wavenumber is involved in the Helmholtz problem, high wavenumber hasto cause the coefficient matrix corresponding to the Helmholtz problem highlyindefinite and non-hermite, and the efficient results of solving this problem cannot begained only by easily using iterative method.With considering the above difficulties, combine preconditioning with Krylovsubspace iterative method for solving the Helmholtz problem in order to get thesatisfied results, which is best when they are independent of wavenumber. Mainlyconsider aggregation-based algebraic multigrid as preconditioning to get thepreconditioner, and then employ Krylov subspace iterative method to solve thepreconditioned system of Helmholtz problem.Apply double pairwise aggregation to the setup process of algebraic multigrid tosolve Helmholtz problem, where double pairwise aggregation scheme was firstlyproposed by Y. Notay in [Aggregation-based algebraic multilevel preconditioning,SIAM J. Matrix Anal. Appl.,2006,27:998–1018]. Compare it with the smoothedaggregation algebraic multigrid. Meanwhile, show shifted Laplacian preconditioners,and observe the influence of the chosen different shift and determine the optimal shift.Also present the condition number and the spectral pictures, namely the distribution ofeigenvalues, of system precondtioned by the three preconditionings for observe whichpreconditioning is best combined with Krylov subspace iterative method. In addition，consider the influence of second-order and fourth-order accurate definite differencediscretization methods on the three techniques.Finally, according to numerical results, find that double pairwise aggregationscheme is a good choice in algebraic multigrid for Helmholtz equations in reducing timeand memory, with the result that the number of iterations and time of solving are mostlyindependent of wavenumber and the results on second-order accurate discretization aremore stable.