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Preconditioners for spectral discretizations of Helmholtz’s equation with Sommerfeld boundary conditions. (English) Zbl 0988.65109
Summary: Some preconditioners for the iterative solution of Helmholtz’s equation discretized with spectral Legendre collocation methods are introduced and studied. The preconditioners are based either on a finite element discretization of Helmholtz’s 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 \(\kappa\).

MSC:
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F35 Numerical computation of matrix norms, conditioning, scaling
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