We consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann, and\nSommerfeld-like boundary conditions based on a compact sixth order approximation scheme and lower order preconditioned\nKrylov subspace methodology. The resulting systems of finite-difference equations are solved by different preconditioned Krylov\nsubspace-based methods. In the analysis of the lower order preconditioning developed here, we introduce the term ââ?¬Å?kth order preconditioned\nmatrixââ?¬Â in addition to the commonly used ââ?¬Å?an optimal preconditioner.ââ?¬Â The necessity of the new criterion is justified\nby the fact that the condition number of the preconditioned matrix in some of our test problems improves with the decrease of\nthe grid step size. In a simple 1D case, we are able to prove this analytically. This new parameter could serve as a guide in the\nconstruction of new preconditioners. The lower order direct preconditioner used in our algorithms is based on a combination\nof the separation of variables technique and fast Fourier transform (FFT) type methods. The resulting numerical methods allow\nefficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative approach.
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