Analysis Of Spectral Galerkin Approximations To The 3D Helmholtz Equation In An Exterior Domain With A Spherical Boundary

Time harmonic wave propagations appear in many applications for Physics and Material sciences as well as Computational Electromagnetics, including wave scatter-ing and transmission, noise reduction, fluid-solid interaction, as well as earthquake wave propagation. Time harmonic wave scattering problems in unbounded domains can be first reduced to a problem on a bounded sphere via the exact Dirichlet-to-Neumann operator, and then an iterative process is presented by applying a kind of local operators, which are chosen suitably to make iterative sequences converge. As it is well known, spectral methods have the main characteristics of high accuracy. When a spectral-Galerkin method is employed to approximate the iterative prob-lem, stability and a total error analysis in weighted Sobolev spaces are presented for the spectral-Galerkin method to the Helmholtz equation in 3D exterior domains. These estimates get rid of an extra factor k1/3 that cannot be removed in the error estimates and a priori bounds of numerical solution of [11]. The estimations appear to have an optimal explicit dependency on the wave number. In addition, we use alternatively two groups of collocation points by spectral collocation method solving nonlinear Burgers equation, and compare numerical errors of those points for small coefficients.