Mixed Spectral Method For Three Dimensional Exterior Problems

Many problems in science and engineering are set in unbound domains, such as fluid dynamics, electromagnetics, quantum mechanics, astrophysics, earth science, financial mathematics and so on. The simplest method to deal with them is to set some artificial boundaries, impose certain artificial boundary conditions and then resolve them numerically. whereas these treatments may cause additional errors. Thus it seems better to solve such problems directly.The main purpose of this work is to develop the mixed spectral method for three-dimensional exterior problems, using spherical harmonic and generalized Laguerre functions. Firstly, we recall some basic properties of generalized Laguerre polynomials and establish some basic approximation results of generalized Laguerre functions. Then, we investigate the mixed approximation using spherical harmonic and generalized Laguerre functions. Next, we propose the mixed spectral schemes for two model problems. Their convergences are proved. Finally, we present some numerical results which demonstrate the spectral accuracy of this new approach.