A Hierarchical 3-D Poisson Modified Fourier Solver by Domain Decomposition
Journal of Scientific Computing
A Modified Fourier–Galerkin Method for the Poisson and Helmholtz Equations
Journal of Scientific Computing
A Fast Spectral Subtractional Solver for Elliptic Equations
Journal of Scientific Computing
A Fourier-Wachspress method for solving Helmholtz's equation in three-dimensional layered domains
Journal of Computational Physics
A Fourier-Wachspress method for solving Helmholtz's equation in three-dimensional layered domains
Journal of Computational Physics
The Trefftz method for the Helmholtz equation with degeneracy
Applied Numerical Mathematics
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We present a fast solver for the Helmholtz equation $$\Delta u \pm \lambda^2 u = f,$$ in a 3D rectangular box. The method is based on the application of the discrete Fourier transform accompanied by a subtraction technique which allows us to reduce the errors associated with the Gibbs phenomenon and achieve any prescribed rate of convergence. The algorithm requires O(N3 log N) operations, where N is the number of grid points in each direction. We solve a Dirichlet boundary problem for the Helmholtz equation. We also extend the method 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 number of points. For example, for the accuracy of 10-8 the resolution of only 16--32 points in each direction is necessary.