Galerkin–Legendre Spectral Method for the 3D Helmholtz Equation

A Galerkin–Legendre spectral method for the direct solution of Poisson and Helmholtz equations in a three-dimensional rectangular domain is presented. The method extends Jie Shenʹs algorithm for 2D problems by using the diagonalization of the three mass matrices in the three spatial directions and fully exploits the direct product nature of the spectral approximation. The Dirichlet boundary values are taken into account by means of a discrete lifting performed in three subsequent steps and built upon Gauss–Legendre quadrature points. A few numerical tests illustrate the accuracy and efficiency of the method.