Advances in Iterative Methods and Preconditioners for the Helmholtz Equation

Abstract

In this paper we survey the development of fast iterative solvers aimed at solving 2D/3D Helmholtz problems. In the first half of the paper, a survey on some recently developed methods is given. The second half of the paper focuses on the development of the shifted Laplacian preconditioner used to accelerate the convergence of Krylov subspace methods applied to the Helmholtz equation. Numerical examples are given for some difficult problems, which had not been solved iteratively before.

Is the Helmholtz Equation Really Sign-Indefinite?

“…Finally, consider the linear system: this is sign-indefinite (as in the Dirichlet case) and nonHermitian (because the boundary condition involves the imaginary unit "i", and therefore a I (u, v) = a I (v, u)); thus the eigenvalues are complex and lie on both sides of the imaginary axis. These facts are not the only reasons why it is difficult to solve the linear systems associated with Helmholtz problems, but they contribute strongly to this difficulty; see the reviews [34], [35], [33], [1] and the references therein for more details.…”

Multigrid methods for Helmholtz problems: A convergent scheme in 1D using standard components

“…Essentially, this means that the coarsest grids employed in the multigrid hierarchy must resolve the waves of wave number k. This is however not very useful in practical situations, particularly for large-scale calculations, since then the finest grid is precisely chosen to resolve the waves of wave number k, and one could not use any coarsening at all in a multigrid context. Thus, it is still the case that classical multigrid methods do not work when applied to discretizations of the Helmholtz equation, see for example [20,22] and references therein. Textbooks on multigrid often comment on the difficulty of the case of indefinite problems, for example the early multigrid guide by Brandt and Livne from 1984, which has recently been republished in the SIAM Classics in Applied Mathematics series [10, page 72], contains the quote: "If σ is negative, the situation is much worse, whatever the boundary con-1 This paper is exemplary for a modern paper in numerical analysis: there are several physical problems modeled by PDEs, which are then discretized, and solved by iteration (Richardson iteration, which is almost the Chebyshev semi-iterative method in its original form, albeit constructing an approximate Chebyshev polynomial by trial and error, and Richardson extrapolation), with realistic wallclock times (flops Richardson and his boy calculators were able to attain), computing costs (…”

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

“…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…”