Summary: In 1983, a preconditioner was proposed by A. Bayliss, C. I. Goldstein, and E. Turkel [J. Comput. Phys. 49, 443–457 (1983; Zbl 0524.65068)] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. A. L. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, St. Hugh’s College, Oxford (2001)] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign.
Both approaches are further generalized to a new class of preconditioners, the so-called “shifted Laplace" preconditioners of the form with . Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.