Fast Solution Methods

TL;DR: 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 and strategies to reduce this complexity are discussed.

Abstract: The standard boundary element method applied to the time harmonic Helmholtz equation yields a numerical method with O(N3) 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 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.