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A preconditioned method for the solution of the Robbins problem for the Helmholtz equation. (English) Zbl 1234.65037

An efficient preconditioned iterative method for the numerical solution of the two dimensional Helmholtz equation with Robbins type boundary conditions is considered. The problem is discretized using an ordinary second order approximation finite difference scheme on a uniform mesh. The matrix of the obtained large sparse system of linear equations is represented by tensor products and tridiagonalized by means of a fast sine transform based preconditioner matrix. The resulting system is solved by the generalized minimum residual method within a finite small number of iterations in exact arithmetic. Spectral analysis of the preconditioned matrix is derived. Convergence rate and operational cost of the method are also discussed. Numerical results illustrate the application of the method.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F50 Computational methods for sparse matrices
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