# On the Minimal Shift in the Shifted Laplacian Preconditioner for Multigrid to Work

## Abstract

Over the past years, the shifted Laplacian has been advocated as a way of making multigrid work for the indefinite Helmholtz equation. The idea is to use a shift into the complex plane of the wave number in the operator, and then to use the shifted operator as a preconditioner for a Krylov method. The hope is that due to the shift, it becomes possible to use standard multigrid to invert the preconditioner, and if the shift is not too big, it is still an effective preconditioner for the Helmholtz equation with a real wave number. There are however two conflicting requirements here: the shift should be not too large for the shifted preconditioner to be a good preconditioner, and it should be large enough for multigrid to work. It was rigorously proved last year that the preconditioner is good if the shift is at most of the size of the wavenumber. We prove here rigorously that if the shift is less than the size of the wavenumber squared, multigrid will not work. It is therefore not possible to solve the shifted Laplace preconditioner with multigrid in the regime where it is a good preconditioner.

## References

- 1.T. Airaksinen, E. Heikkola, A. Pennanen, J. Toivanen, An algebraic multigrid based shifted-laplacian preconditioner for the Helmholtz equation. J. Comput. Phys.
**226**(1), 1196–1210 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 2.A. Bayliss, C. Goldstein, E. Turkel, An iterative method for the Helmholtz equation. J. Comput. Phys.
**49**, 443–457 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 3.A. Brandt, O.E. Livne, in
*Multigrid Techniques, 1984 Guide with Applications to Fluid Dynamics*, Revised Edition. Classics in Applied Mathematics, vol. 67 (SIAM, Philadelphia, 2011)Google Scholar - 4.S. Cools, W. Vanroose, Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems. Numerical Linear Algebra with Applications
**19**(2), 232–252 (2013)MathSciNetzbMATHGoogle Scholar - 5.Y. Erlangga, Advances in iterative methods and preconditioners for the Helmholtz equation. Arch. Comput. Meth. Eng.
**15**, 37–66 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Y. Erlangga, C. Vuik, C. Oosterlee, On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math.
**50**, 409–425 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 7.O. Ernst, M. Gander, Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods, in
*Numerical Analysis of Multiscale Problems*, ed. by I. Graham, T. Hou, O. Lakkis, R. Scheichl (Springer, Berlin, 2012), pp. 325–363CrossRefGoogle Scholar - 8.M. Gander, O. Ernst, Multigrid Methods for Helmholtz Problems: A Convergent Scheme in 1d Using Standard Components, in
*Direct and Inverse Problems in Wave Propagation and Applications*(De Gruyter, Boston, 2013), pp. 135–186zbMATHGoogle Scholar - 9.M. Gander, I.G. Graham, E.A. Spence, How should one choose the shift for the shifted laplacian to be a good preconditioner for the Helmholtz equation? Numer. Math. (2015). doi:10.1007/s00211-015-0700-2Google Scholar
- 10.M.V. Gijzen, Y. Erlangga, C. Vuik, Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput.
**29**(5), 1942–1958 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 11.W. Hackbusch,
*Multi-Grid Methods and Applications*(Springer, Berlin, 1985)CrossRefzbMATHGoogle Scholar - 12.U. Trottenberg, C.C.W. Oosterlee, A. Schüller,
*Multigrid*(Academic Press, New York, 2001)zbMATHGoogle Scholar