The standard boundary element method applied to the time harmonic Helmholtz equation yields a numerical method with O(N 3 ) complexity when using a direct solution of the fully populated system of linear equations. Strategies to reduce this complexity are discussed in this paper. The O(N 3 ) complexity issuing from the direct solution is first reduced to O(N 2 ) by using iterative solvers. Krylov subspace methods as well as strategies of preconditioning are reviewed. Based on numerical examples

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... numerical examples the influence of different parameters on the convergence behavior of the iterative solvers is investigated. It is shown that preconditioned Krylov subspace methods yields a boundary element method of O(N 2 ) complexity. A further advantage of these iterative solvers is that they do not require the dense matrix to be set up. Only matrix-vector products need to be evaluated which can be done efficiently using a multilevel fast multipole method. Based on real life problems it is shown that the computational complexity of the boundary element method can be reduced to O(N log 2 N ) for a problem with N unknowns.