Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

Abstract

There has been much recent research on preconditioning discretisations of the Helmholtz operator + k 2 (subject to suitable boundary conditions) using a discrete version of the so-called "shifted Laplacian" + (k 2 + iε) for some ε > 0. This is motivated by the fact that, as ε increases, the shifted problem becomes easier to solve iteratively. Despite many numerical investigations, there has been no rigorous analysis of how to chose the shift. In this paper, we focus on the question of how large ε can be so that the shifted problem provides a preconditioner that leads to k-independent convergence of GMRES, and our main result is a sufficient condition on ε for this property to hold. This result holds for finite element discretisations of both the interior impedance problem and the sound-soft scattering problem (with the radiation condition in the latter problem imposed as a far-field impedance boundary condition). Note that we do not address the important question of how large ε should be so that the preconditioner can easily be inverted by standard iterative methods.Mathematics Subject Classification 35J05 · 65F08 · 65F10 · 65N30 · 78A45

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

“…the value of β 2 ) needs to balance the quality of the preconditioner (favoring small damping) with the ease to invert it (favoring large damping). This balance has received considerable attention in the literature [16,11,17,29,21]. The results of the recent paper [21] indicate that the complex shift in (3.1) should scale like the wave number k in order to obtain an optimal preconditioner for the Helmholtz operator.…”

“…This balance has received considerable attention in the literature [16,11,17,29,21]. The results of the recent paper [21] indicate that the complex shift in (3.1) should scale like the wave number k in order to obtain an optimal preconditioner for the Helmholtz operator. Results in the paper [29] instead indicate that the combined use of the CSLP preconditioner allows to increase the amount of damping without compromising the convergence of the outer Krylov iteration.…”

“…The idea of adding weight to the diagonal of the ILU preconditioner forms the basis of [14]. The CSLP preconditioners were further developed in [15,16] and later generalized in [17,18,19,20,21]. This lead to a breakthrough in industrial applications [22,23,24].…”

Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the $$hp$$ h p -Version

“…Proof For the estimate (6), we refer the reader to [12,Theorem 2.18], [33,Sect. 4] and, in particular [33,Remark 4.7].…”

An efficient multigrid algorithm for heterogeneous acoustic media sign‐indefinite high‐order FEM models

“…Mathematical analysis of how to choose the shift parameter for the homogeneous media model was investigated only recently in , where the generalized minimal residual method (GMRES) is applied to iteratively solve the discrete Helmholtz algebraic system. However, the analysis in does not take into account the use of multigrid methods for efficiently constructing the shifted Laplacian preconditioner. The investigation in suggests that it is efficient to choose the shift to be proportional to the wavenumber k .…”