Preconditioners for spectral discretizations of Helmholtz's equation with Sommerfeld boundary conditions

Some preconditioners for the iterative solution of Helmholtz's equation dis cretized with spectral Legndre collocation methods are introduced and studi ed. The preconditioners are based either on a finite element discretization of Helmholtz's equation on the spectral collocation mesh or on replacing t he Sommerfield-like boundary condition on a subset of the boundary with eit her Neumann or Dirichlet boundary conditions. The convergence rate of the r esulting iterative methods is only mildly dependent on the spectral degree N and the wave number kappa. (C) 2001 Elsevier Science B.V. All rights rese rved.