Spectral Element Method For Inhomogeneous Boundary Value Problems Of Fourth-order And Domain Decomposition Spectral Method For Second-order Exterior Problem With Arbitrary Polygonal Obstacle

During the past thirty years, spectral and pseudospectral methods, as importanttools for numerical solutions of differential equations, developed rapidly. Theypossess the high accuracy, and so have been applied successfully to numericalsimulations of practical problems arising in many fields, such as fluid mechanics,quantum mechanics, statistical physics, weather prediction, marine science, chemicalreactions, material science, bio-engineering, astrophysics, financial mathematics andso on. The usual spectral and pseudospectral methods are only available for periodicproblems, and problems defined on rectangular domains. Some authors proposedthe Jacobi and generalized Jacobi orthogonal approximations, and the Jacobi-Gausstype interpolations, as well as the related spectral and pseudospectral methods fordegenerated, singular and higher order problems. Recently, several authors alsodeveloped the one and two-dimensional Legendre quasi-orthogonal approximationsand the corresponding interpolations, which opened a new goal for the domaindecomposition spectral and pseudospectral methods, and the spectral andpseudospectral element methods.Theoretically, the larger the modes used in spectral approximations, the smallerthe numerical errors. However, it is not convenient to take vary large modes in actualcomputations. In opposite, the spectral and pseudospectral element methods, forwhich we divide considered domains into several subdomains and use differentapproximations on different subdomains. As results, they simplify the computationsand raise the numerical accuracy. Therefore, they become one of the mostlyadvanced topic in the study of spectral and pseudospectral methods. There are threedifficulties in designing and analyzing such methods. The first one is how to matchnumerical solutions and their derivatives on all interfaces of adjacent subdomains,and how to ensure the global spectral accuracy. The second one is how to analyzethe spectral accuracy of numerical solutions on the whole domains. Thus, we neednew framework in numerical analysis. The third one is how to design efficient algorithms. In other words, we should construct specific basis functionscorresponding to various subdomains and their edges and vertices. An important andunsettled problem is the spectral element method for mixed inhomogeneousboundary value problems of high-order partial differential equations, for which wenot only have to keep the continuity of approximate functions and their derivativeson all interfaces of adjacent subdomains, but also to maintain the global spectralaccuracy of composite approximations on the whole domains.In this thesis, we investigate the spectral element method for mixedinhomogeneous boundary value problems of fourth-order partial differentialequations. To do this, we first propose a new Legendre quasi-orthogonalapproximation in rectangles with the best order error estimates. Secondly, wedevelop the composite Legendre quasi-orthogonal approximation on the wholedomains, which could be divided into some rectangles. This approximation serves asthe mathematical foundation of spectral element method for mixed inhomogeneousboundary value problems of fourth-order. Moreover, we design proper basisfunctions in actual computations, corresponding to subdomains and their edges andvertices. Such basis functions and their derivatives are continuous on all interfaces ofadjacent subdomains. Furthermore, we provide the spectral element scheme forfourth-order problem with mix inhomogeneous boundary conditions. Since we adoptthe different quasi-orthogonal approximations on the different subdomains and theiredges,the suggested new approach is a high order method with non-uniform meshesand non-uniform modes. Accordingly, it is specially appreciate for problems withsolutions behaving differently on different subdomains.Nowadays, the spectral and pseudospectral methods for solving differentialequations defined on unbounded domains are also developing rapidly. As we known,we oftentimes need to solve numerically exteral problems arising in fluid mechanics,electromagnetic theory, ecology and so on. The early work focused on exteriorproblems with disk and ball obstacles by using the Laguerre-Fourier and Laguerre-Spherical harmonic approximations. A more important and challengingsubject is how to deal with external problems with other kinds of obstacles. Weusually take a rectangle containing the obstacle, and divide the considered externaldomain into two parts. The first part is the area between the boundaries of theobstacle and the rectangle. The other one is the remaining unbounded subdomain.However, we face three difficulties in designing and analyzing the related spectralmethods. The first one is how to approximate the underlying problems on theunbounded subdomain. The second one is how to match numerical solutions on theinterfaces of adjacent subdomains. The third one is how to ensure the global spectralaccuracy. For the boundary value and initial-boundary value problems with arbitrarypolygonal obstacles, a special difficulty is how to match numerical solutions on allinterfaces between the unbounded subdomains and the bounded subdomains. Inother words, we are obliged to consider certain specific quasi-orthogonalapproximations and interpolations on the bounded subdomains and unboundedsubdomains, so that the corresponding composite quasi-orthogonal approximationsand interpolations on the whole exterior domains are not only continuous on theinterfaces, but also keep the global spectral accuracy. Besides, we should designreasonable basis functions such that the numerical solutions keep the continuity onall interfaces. Moreover, the induced spectral and pseudospectral schemes should beeasy to be implemented numerically.As the second topic of this thesis, we study domain decomposition spectralmethod for exterior problems with arbitrary polygonal obstacles. For this purpose,we first propose the new Legendre quasi-orthogonal approximation on quadrilaterals,the Legendre-Laguerre quasi-orthogonal approximation on infinite strips and thetwo-dimensional Laguerre quasi-orthogonal approximation on the quarters, andestablish the corresponding basic error estimates with the best order, respectively.Next, we build up the composite quasi-orthogonal approximation on the wholeexterior domains, which not only keeps the continuity of approximate functions onall interfaces of adjacent subdomains, but also possesses the global spectral accuracy on the whole domains. Consequently, it serves as the mathematical foundation ofdomain decomposition spectral method for second-order exterior problems withinhomogeneous boundary conditions. Moreover, we design the reasonable basisfunctions corresponding to different kinds of subdomains and their edges andvertices, keeping the continuity on all interface of adjacent subdomains. Accordingly,we provide the domain decomposition spectral scheme for exterior problems witharbitrary polygonal obstacles. In particular, we propose a spectral method withnon-uniform modes for problems with solutions changing very rapidly or oscillatingseriously somewhere. This treatment saves the work and raises the numericalaccuracy.