A Fully Diagonalized Generalized Laguerre Spectral Method For Unbounded Domains

During the past forty years, significant progress has been made in the research on spectral methods. They play an important role in scientific and engineering computing and have been applied successfully to numerical simulations in many fields, such as heat conduction, quantum mechanics, fluid dynamics, meteorology, finance mathematics and so on. Nowadays, spectral methods, as well as finite element and finite difference methods, are some of the powerful tools for numerical solutions of partial differential equations.The fascinating merit of spectral methods is the high accuracy, so-called convergence of infinite order. The smoother the genuine solution, the higher the convergence rate of the numerical solution by spectral methods. Particularly, when the genuine solution is infinitely differentiable, the numerical solution converges faster. The remarkable fea-ture of spectral methods is to choose various orthogonal systems of infinitely differentiable functions as trial functions. Different trial functions lead to different spectral approxima-tions. For instance, trigonometric polynomials for periodic problems, Jacobi polynomials for non-periodic problems, Laguerre polynomials and Hermite polynomials for problems on unbounded domains.Generally, spectral methods for solving partial differential equations on unbounded domains can be essentially classified into three catalogues:(ⅰ). truncate an unbounded domain to a bounded one and solve the problem on the bounded domain subject to artificial or transparent boundary conditions; (ⅱ). map the original problem on an unbounded domain to one on a bounded domain and use classic spectral methods to solve the new problem; or equivalently, approximate the original problem by some non-classical functions mapped from the classic orthogonal polynomials/functions on a bounded domain; (ⅲ). directly approximate the original problem by genuine orthogonal functions such as Laguerre polynomials or functions on the unbounded domain.The third approach is of particular interest to researchers, and has won an increasing popularity in a broad class of applications, owing to its essential advantages over other two approaches. These direct approximation schemes constitute an initial step towards the efficient spectral methods, which admit fast and stable algorithms for their efficient implementations. However, the classical Laguerre spectral methods can not be used to solve some problems for the parameter α>- 1. Based on this reason, we propose the generalized Laguerre spectral methods with any real parameter a, establish the fully diagonalized Laguerre spectral methods to solve some elliptic boundary value problems on unbounded domains. The fascinating merit of this method is the resulting algebraic systems is a unit matrices and all the condition numbers of the corresponding total stiff matrices are equal to 1. For the classic Laguerre spectral methods, the corresponding total stiff matrices have off-diagonal entries, the condition numbers of the resulting systems increase asymptotically as O(N2).The construction of this paper is organized as follows:In chapter 1, we recall the history of spectral methods. We describe the motivation, the difficulties and the main results of this paper.In chapter 2, we introduce generalized Laguerre polynomials and functions with arbi-trary index α. We establish such a unified orthogonal Laguerre projection and obtain the convergence analysis of this projection.In chapter 3, the fully diagonalized Laguerre spectral methods and the implementa-tion of algorithms are proposed for the Dirichlet and Robin boundary value problems of second order elliptic equations. Numerical results are presented to demonstrate the effec-tiveness and accuracy of the proposed diagonalized Laguerre spectral methods, which are in agreement with our theoretical predictions.In chapter 4, we propose the fully diagonalized Laguerre spectral methods and the implementation of algorithms for the second-order symmetric elliptic problems under polar (resp. spherical) coordinates in R2 (resp. R3). We also analyze the numerical errors and present some numerical results for the suggested diagonalized Laguerre spectral methods of second-order elliptic problems.In chapter 5, we devote to construct the fully diagonalized Laguerre spectral methods and the implementation of algorithms for the fourth-order elliptic boundary value problems on the half line. The convergence analysis and numerical results of fourth-order elliptic problems are also presented to demonstrate the effectiveness and accuracy of the proposed diagonalized Laguerre spectral methods. Next, we propose the fully diagonalized Laguerre spectral methods and the implementation of algorithms for the second-order symmetric elliptic exterior problems in R2 (resp. R3). We also present some numerical results for the suggested diagonalized Laguerre spectral methods of second-order exterior problems.