 
Summary: A FAST SPECTRAL SOLVER FOR 3D HELMHOLTZ EQUATION
A. AVERBUCH y , E. BRAVERMAN z , AND M. ISRAELI x
Abstract. We present a fast solver for the Helmholtz equation
\Deltau \Sigma – 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 to reduce the errors associated with the Gibbs
phenomenon and achieve any prescribed rate of convergence. The algorithm requires O(N 3 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 on some faces Dirichlet boundary conditions are specified and Neumann boundary conditions
on the other faces. High order accuracy is achieved by a comparatively small number of points. For
example, for the accuracy of 10 \Gamma8 the resolution of only 1632 points in each direction is necessary.
Key words. Fast 3D solver, Helmholtz equation, Fourier method, Dirichlet and mixed bound
ary conditions
AMS subject classifications. 65N35, 65T20, 35J05
1. Introduction. A fast and accurate solution of elliptic equations is often an
important step in the process of solving problems of fluid dynamics and other scien
tific computing applications. Helmholtz type equations usually appear in acoustics
or electromagnetics and also as a result of time discretization of the NavierStokes
equations.
