Preconditioning of complex linear systems from the Helmholtz equation

In this paper, for solving a class of complex linear systems from the Helmholtz equation efficiently, a new splitting preconditioner is established and a real-valued preconditioned iterative method is presented. Spectral properties of the preconditioned matrix are discussed and bound on the eigenvalues of the preconditioned matrix is given. Theorem which provides the dimension of the Krylov subspace methods for the preconditioned iteration method is obtained. The implementation of the preconditioned method is given and the optimal choice of the accelerated parameter is derived. In particular, a more practical way to choose the accelerated parameter is also proposed. Numerical experiments arising from the Helmholtz equation are used to illustrate the performance of the preconditioner, which show the effectiveness and robustness of the new preconditioned GMRES method and demonstrate meshsize-independent and wave number-insensitive convergence behavior.