Preconditioners for spectral discretizations of Helmholtzʹs equation with Sommerfeld boundary conditions Original Research Article

Some preconditioners for the iterative solution of Helmholtzʹs equation discretized with spectral Legendre collocation methods are introduced and studied. The preconditioners are based either on a finite element discretization of Helmholtzʹs equation on the spectral collocation mesh or on replacing the Sommerfeld-like boundary condition on a subset of the boundary with either Neumann or Dirichlet boundary conditions. The convergence rate of the resulting iterative methods is only mildly dependent on the spectral degree N and the wave number κ.