A fast spectral solver for a 3D Helmholtz equation

We present a fast solver for the Helmholtz equation Delta u +/- lambda(2) u = f; in a 3D rectangular box. The method is based on the application of the disc rete Fourier transform accompanied by a subtraction technique which allows us to reduce the errors associated with the Gibbs phenomenon and achieve an y prescribed rate of convergence. The algorithm requires O(N-3 log N) opera tions, where N is the number of grid points in each direction. We solve a D irichlet boundary problem for the Helmholtz equation. We also extend the me thod to the solution of mixed problems, where Dirichlet boundary conditions are specified on some faces and Neumann boundary conditions are specified on other faces. High-order accuracy is achieved by a comparatively small nu mber of points. For example, for the accuracy of 10(-8) the resolution of o nly 16-32 points in each direction is necessary.