Solving the discretized Helmholtz equation on large scale remains to date a challenging problem. The idea of using the complex shifted Laplacian as a preconditioner for an outer Krylov subspace iteration was introduced in [1]. This paper constitutes a landmark in the development of fast solution algorithms as can be judged from the number of citations. The combination of the complex shifted Laplacian with a multigrid deflation technique was first proposed in [2] and later analyzed in [3]. In these works the shifted Laplacian is attributed the role of a multigrid smoother. Fourier analysis and numerical results show that while the use of a coarse grid correction type iteration does bring the required number of iterations down, the resulting algorithm fails to be scalable especially at high wave number values. Indeed, the linear interpolation employed fails to sufficiently remove the near-null space components of the kernel. Inspired by this work we consider other ways to remove the error components that are slow to converge for the shifted Laplacian. We show that a V-cycle that converges for the high energy modes can be build and exploit this fact to expose the near-null space modes. This modes are used subsequently to build an effective deflation operator. Numerical results will illustrate the effectiveness of our approach.
[1] Y. A. Erlangga, C. Vuik and C.W. Oosterlee, On a Class of Preconditioners for Solving the Helmholtz Equation, Appl. Numer. Math., pp. 409-425, 50, (3-4), 2004.
[2] Y. A. Erlangga and R. Nabben, On a Multilevel Krylov Method for the Helmholtz Equation Preconditioned by Shifted Laplacian, ETNA, pp. 403-424, 31, 2008.
[3] A. Sheikh, D. Lahaye and C. Vuik, On the Convergence of the Shifted Laplace Preconditioner Combined with Multigrid Deflation, submitted to NLAA.