Mixed Spectral And Pseudospectral Methods For Exterior Problems

Many practical problems arising in science and engineering are set on exterior domains,suchas some problems with obstacles in ?uid dynamics and so on. The simplest method to deal withthem is to set some artificial boundaries, impose certain artificial boundary conditions and thenresolve them numerically, whereas these treatments may cause additional errors. Thus it seemsreasonable to solve such problems directly.The main purpose of this work is to develop the mixed spectral methods for exterior prob-lems. We first study the mixed spectral and pseudospectral methods for two-dimensional exte-rior problems, using Fourier and generalized Laguerre functions. Then, we propose the mixedpseudospectral method for three-dimensional exterior problems, using spherical harmonic andgeneralized Laguerre functions.This work consists of four parts. In Chapter 1, we recall the history of numerical methods forexterior problems. We describe the motivation, the difficulties and the main results of this work.In Chapter 2, we recall some basic properties of generalized Laguerre functions. We establishsome basic results on the mixed Fourier-Laguerre orthogonal approximation. As examples, wepropose the mixed spectral schemes for two-dimensional exterior problems. The convergences ofproposed schemes are proved. Numerical results demonstrate the efficiency of this approach.In Chapter 3, we establish some basic results on the mixed Fourier-Laguerre interpolation. Asexamples, we propose the mixed pseudospectral schemes for two-dimensional exterior problems.The convergences of proposed schemes are proved. Numerical results show the efficiency of thisapproach. In particular, it simplifies actual computation and saves a lot of computational time.In Chapter 4, we establish some basic results on the mixed spherical harmonic-generalizedLaguerre interpolation, which play important roles in the related pseudospectral methods. As anexample, we provide the mixed pseudospectral scheme for a model problem outside a ball. Theconvergence of proposed scheme is proved. Numerical results demonstrate the spectral accuracyof this approach.