Modern Solvers for Helmholtz Problems pp 85-112 | Cite as

# How to Choose the Shift in the Shifted Laplace Preconditioner for the Helmholtz Equation Combined with Deflation

## Abstract

In recent work we showed that the performance of the complex shifted Laplace preconditioner for the discretized Helmholtz equation can be significantly improved by combining it multiplicatively with a deflation procedure that employs multigrid vectors. In this chapter we argue that in this combination the preconditioner improves the convergence of the outer Krylov acceleration through a new mechanism. This mechanism allows for a much larger damping and facilitates the approximate solve with the preconditioner. The convergence of the outer Krylov acceleration is not significantly delayed and occasionally even accelerated. To provide a basis for these claims, we analyze for a one-dimensional problem a two-level variant of the method in which the preconditioner is applied after deflation and in which both the preconditioner and the coarse grid problem are inverted exactly. We show that in case that the mesh is sufficiently fine to resolve the wave length, the spectrum after deflation consists of a cluster surrounded by two tails that extend in both directions along the real axis. The action of the inverse of the preconditioner is to shrink the length of the tails while at the same time rotating them and shifting the center of the cluster towards the origin. A much larger damping parameter than in algorithms without deflation can be used.

## Keywords

Grid Point Dirichlet Boundary Condition Coarse Grid Helmholtz Equation Krylov Subspace## Notes

### Acknowledgements

The authors would like to sincerely thank Prof. Ira Livshits and Prof. Martin Gander for numerous discussions that indirectly give raise to material in this paper.

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