On a class of preconditioners for solving the Helmholtz equation

Abstract

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] 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. 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. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called "shifted Laplace" preconditioners of the form ∆φ − αk 2 φ with α ∈ C. Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.

Iterative schemes for high order compact discretizations to the exterior Helmholtz equation

“…This convergence degradation when k increases, is relatively mild, compared to the unpreconditioned Bi-CGSTAB. The latter typically shows extremely slow convergence for even small k [8,10] and an exponential increase of the number of iterations as k increases.…”

“…An example of this is basing the preconditioner on the solution of the Laplace equation [2]. Erlangga et al [7][8][9][10] extended this idea by considering a Helmholtz equation with a complex wave number as the preconditioner, namely…”

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

“…The use of different approximations for F −1 d have been studied in [30,21,38,34]. Here an algebraic multigrid approximation described in Section 5 is considered.…”

“…with a complex shift z 2 = α 2 + β 2 i was suggested in [30] as a preconditioner for the Helmholtz equation. By choosing α 2 = 1 and β 2 to be negative, F d is the Helmholtz operator in (1) with some additional damping.…”

“…The class of preconditioners based on damped operators that are considered here, is an example of this approach. A shifted-Laplacian preconditioner with a complex shift, which is called here a damped Helmholtz preconditioner, was first considered in [30] for the Helmholtz equation. This was a development over the shifted-Laplacian preconditioner with a real shift previously described in [31].…”