A direct Chebyshev collocation method for the numerical solutions of three-dimensional Helmholtz-type equations

Abstract

In this study, a new framework for the numerical solutions of inhomogeneous Helmholtz-type equations on three-dimensional (3D) arbitrary domains is presented. A Chebyshev collocation scheme (CCS) is introduced for the efficient and accurate approximation of particular solution for the given 3D boundary value problem. We collocate the numerical scheme at the Gauss–Lobatto nodes to ensure the pseudo-spectral convergence of the Chebyshev interpolation. After the particular solution is evaluated, the introduced CCS is coupled with a two-stage and one-stage numerical schemes to evaluate the final solutions of the given problem. In the two-stage approach, the given inhomogeneous problem is converted to a homogeneous equation and then the boundary-type methods, such as the method of fundamental solutions (MFS), can be used to evaluate the resulting homogeneous solutions. In the one-stage scheme, by imposing the boundary conditions directly to the CCS procedure, the final solutions of the given inhomogeneous problem can be obtained straightforward without the need of using the MFS or other boundary methods to find the homogeneous solution. Two benchmark numerical examples in both smooth and piecewise smooth 3D geometries are presented to demonstrate the applicability and efficiency of the proposed method.