Generalized Jacobi Quasi-orthogonal Approximation And Interpolation And Their Applications In The Petrov-Galerkin Spectral, Spectral Element And Collocation Methods For High Order Differential Equations

During the past thirty years, spectral and pseudospectral methods developed rapidly. They possess the high accuracy, and serve as one of important tools for scientific computing. The traditional spectral methods are available for periodic problems and various problems defined on rectangular domains, by taking trigonometric polynomial, Legendre polynomial or Chebysheve polynomial as the basis functions. They have been used widely for boundary and initial-boundary value problems of second and fourth order differential equations. In actual computations, pseudospetral and collocation methods are more preferable sometimes, with which we only need to evaluate unknown functions on certain interpolation nodes, and could deal with nonlinear problems more easily. The mathematical foundation of these approaches are the Legendre-Gauss type and the Chebyshev-Gauss type interpolations.Some authors developed the Jacobi orthogonal approximation and the Jacobi-Gauss type interpolation in the weighted Sobolev space, and proposed the Jacobi spectral and pseudospectral methods for degenerated differential equations on bounded domains. They have been also applied to some problems defined on unbounded domains and axisymmetric domains by using certain proper variable transformations. The Jacobi orthogonal approximation is related closely to the spectral method on triangles, as well as various rational and irrational spectral methods for unbounded domains. We usually consider second and fourth order differential equations. But, it is also important to consider numerical solutions of high order problems. Recently, some authors proposed the generalized Jacobi orthogonal approximation, which leads to the generalized Jacobi spectral method for high order differential equations. However, it is only appropriate for high order differential equations with homogeneous Dirichlet boundary condition, and not applicable to domain decomposition spectral method and spectral element method.In this thesis, we study the generalized Jacobi quasi-orthogonal approximation and the related Jacobi-Gauss-Lobatto interpolation in one dimension and two dimensions, which lead to the new generalized Jacobi spectral method, generalized Jacobi spectral element method and collocation method.We propose the generalized Jacobi quasi-orthogonal approximation in one dimension, which matches the approximated functions and some of its derivatives at the endpoints of finite intervals exactly. Moreover, it keeps the same accuracy as the usual orthogonal approximation. Thus, it serves as the mathematical foundation of the Petrov-Galerkin spectral and spectral element methods for one-dimensional high order differential equations with mixed nonhomogeneous boundary and initial-boundary conditions. As some applications, we propose the Petrov-Galerkin spectral method for a odd and high order differential equation, and the Petrov-Galerkin spectral element method for a mixed nonhomogeneous Dirichlet-Neumann boundary value problem of fourth order differential equation, numerical results demonstrate the efficiency of suggested algorithms.Next, we investigate the generalized Jacobi-Gauss-Lobatto interpolation in one dimension. Such interpolation matches the approximated function and some of its derivatives at the endpoints of finite intervals, and so induces new pseudospectral method for high order differential equations in one dimension. As an application, we consider the collocation method for the nonlinear Klein-Gordon equation.Finally, we study the generalized Jacobi orthogonal approximation and the related Jacobi-Gauss-Lobatto interpolation in two dimensions. They induce new spectral and pseudospectral methods, which can be applied to the numerical solutions of singular and degenerated differential equations of high order. As examples of applications, we design spectral schemes for two singular and degenerated partial differential equations of fourth order.