| Spectral methods developed rapidly in the past three decades, which serve as importanttools for solving di?erential equations numerically. The fascinating merit of spectral methodsis their high accuracy. Thereby, they have been applied successfully to numerical simulationsin many fields, such as ?uid dynamics, quantum mechanics, meteorology and so on. Theusual spectral methods are only available for periodic or semi-periodic problems, and prob-lems defined on rectangular domains. However, many practical problems arising in scienceand engineering require solving partial di?erential equations defined on unbounded domainsor exterior problems. One of methods for solving such problems is to set certain artificialboundaries, impose some artificial boundary conditions and then resolve the correspondingapproximated problems. Whereas these treatments cause additional errors. Thus it seemsbetter to solve them directly.The purpose of this work is to develop the composite Laguerre-Legendre spectral andpseudospectral methods for problems defined on unbounded domains and exterior problems.Their mathematical foundation are orthogonal approximations and interpolations by takingcertain orthogonal polynomials and functions as base functions, coupled with domain decom-position.This work consists of four parts. In Chapter 1, we recall the history of numerical methodsfor unbounded domains brie?y, in particular, the spectral methods for unbounded domainsand exterior problems. We describe the motivation, the di?culties and the main results ofthis work.In Chapter 2, we recall some results on the one-dimensional Laguerre approximation, andthe one-dimensional Legendre approximation. We shall take the Laguerre functions with a pa-rameterβand Legendre polynomials as base functions, to approximate underlying problems.By suitable choice of parameterβ, we could fit the exact solutions more properly.In Chapter 3, we study the composite Laguerre-Legendre spectral and pseudospectralmethods for the Fokker-Planck equation in an infinite channel, which is a partial di?eren-tial equation of non-standard type. Some numerical algorithms were proposed for solving this equation. Tang, Mckee and Reeks [60] used the Hermite spectral method for a simpli-fied one-dimensional model which is a standard parabolic equation. Recently, Fok, Guo andTang [12] proposed the mixed Hermite spectral-finite di?erence scheme for two-dimensionalproblem. Whereas the numerical accuracy is limited. An interesting and challenging prob-lem is how to use spectral method for various important partial di?erential equations ofnon-standard types on unbounded domains, such as the Fokker-Planck equation, for obtain-ing numerical solutions with high accuracy. The di?culties of dealing with this problemnumerically are caused by several facts. Firstly, it behaves like parabolic equation in onedirection, and behaves like hyperbolic equation in another direction. Thereby, we could notdesign the spectral schemes and analyze the numerical errors in the usual way. Next, thesolution satisfies di?erent kinds of boundary conditions on di?erent subdomains. Therefore,we have to use domain decomposition and di?erent approximations on di?erent unboundedsubdomains. Finally, some coe?cients appearing in the Fokker-Planck equation vary from?∞to∞. This mater also brings a lot of di?culties in actual computation and numericalanalysis. To remedy the above di?culties, we first establish some results on the generalizedLaguerre quasi-orthogonal approximation and Legendre quasi-orthogonal approximation inone-dimension. Then, we introduce di?erent mixed Laguerre-Legendre approximations ondi?erent unbounded subdomains, respectively. By using these results with domain decompo-sition, we bulid up the composite Laguerre-Legendre approximation on the whole domain. Asan important application, the composite spectral scheme is provided for the Fokker-Planckequation in an infinite channel. The convergence of proposed scheme is proved, as well asthe spectral accuracy in space. An e?cient algorithm is described. Numerical results showthe high accuracy of this approach and corfirm the analysis well . In order to simplify ac-tual comptation, we also consider the composite Laguerre-Legendre pseudospectral methodfor the Fokker-Planck equation in an infinite channel. To do this, we establish some re-sults on the Laguerre-Gauss-Radau interpolation and Legendre-Gauss-Radau interpolation inone-dimension. Then, we introduce the mixed Laguerre-Legendre-Gauss-Radau interpolationand two-dimension Laguerre-Gauss-Radau interpolation on di?erent unbounded subdomains,respectively. Along with these results, we build up the composite Laguerre-Legendre-Gauss- Radau interpolation on whole domain. With the aid of these results, we propose the compositescheme. An e?cient implementation is provided. Numerical results show the e?ciency of thisapproach. In particular, it simplifies actual computation and saves a lot of computationaltime.In Chapter 4, we investigate the composite Laguerre-Legendre spectral and pseudospec-tral methods for two-dimensional exterior problems with domain decomposition. Some au-thors developed domain decomposition spectral methods for one dimensional problems, see[25, 56, 57]. Also in [27, 29, 36], spectral methods for exterior problems with a ball obstacleor a disk obstacle were considered. However, the more practical and di?cult problem is howto solve exterior problems with polygon obstacles. There are three di?culties in designingand analyzing the spectral schemes for such problems. The first problem is how to approxi-mate the underlying problems on various unbounded subdomains. The second, but the mostdi?cult one is how to match the numerical solutions on the common boundaries of adjacentsubdomains. The third is how to keep the spectral accuracy on the whole domain and how toestimate the global numerical errors. To remedy these di?culties, we establish some resultson the generalized Laguerre quasi-orthogonal approximation and Legendre quasi-orthogonalapproximation in one-dimension. Then, we turn to the mixed generalized Laguerre-Legendreapproximations and two-dimensional Laguerre approximations on di?erent unbounded sub-domains, respectively. By using these results, we build up the composite Laguerre-Legendreapproximation on the whole exterior domain. Accordingly, we provide the composite spectralschemes for two model problems defined on exterior domains. The convergences of proposedschemes are proved, with the spectral accuracy in space. E?cient algorithms are described.For simplifying actual computation, we introduce two kinds of specific basis functions. Nu-merical results demonstrate the high e?ciency of this new approach and coincide well with theanalysis. Furthermore, we consider the composite pseudospectral method for two-dimensionalexterior problems. We establish some results on the generalized Laguerre-Gauss-Radau in-terpolation approximation and Legendre-Gauss-Lobatto interpolation approximation in one-dimension. Then, we build up the mixed generalized Laguerre-Legendre interpolations andtwo-dimensional Laguerre interpolations on di?erent unbounded subdomains, respectively. With the aid of domain decomposition, they conform the composite Laguerre-Legendre in-terpolation on the whole domain. The composite pseudospectral schemes are provided fortwo model problems defined on exterior domains. The convergences of proposed schemes areproved. E?cient algorithms are described. Numerical results demonstrate the high accuracyof this new approach. They also indicate that the composite pseudospectral method is eas-ier to be implemented and saves much computational time than the corresponding spectralmethod.The approximation results given in this work enrich the spectral methods and enlarge theirapplications, especially the domain decomposition spectral methods. The techniques used inthis paper are also applicable to numerical solutions of many other problems on unboundeddomains, as well as exterior problems.
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