Generalized Jacobi Rational Spectral Methods For Exterior Problems And Neumann 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.Moreover, in a standard variational formulation for Neumann problems, the Neumann bound-ary condition is commonly imposed in a natural way. But this approach usually leads to a fullstiffness matrix for approximating the second derivatives. Therefore, it is more appropriate todevelop a new approach such that a n-diagonal stiffness matrix is employed, instead of the fullstiffness matrix encountered in the classical variational formulation.The main purpose of this work is to develop two new spectral methods, that is:1. the mixed Fourier-generalized Jacobi rational spectral methods for exterior problems;2. the generalized Jacobi rational spectral methods with essential imposition of Neumannboundary conditions for Neumann problems.This work consists of three parts. In Chapter 1, we recall the histories of numerical methodsfor exterior problems and Neumann problems. We also describe the motivation of this work.In Chapter 2, we recall some basic properties of generalized Jacobi rational functions, andestablish some basic results on the mixed Fourier-generalized Jacobi rational orthogonal approxi-mation. As examples, we design the mixed spectral schemes for two-dimensional exterior problem.The convergence of proposed schemes is proved. Especially, taking suitable base functions, the re-sultant linear discrete systems are symmetric and sparse. Thereby, we can resolve them efficiently.Numerical results demonstrate the efficiency of this approach.In Chapter 3, we focus on the Neumann problems on unbounded domains, using the general-ized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions.For analyzing the numerical errors, we establish some basic results on the generalized Jacobi ra-tional approximations for Neumann problems. We also consider a 1-D problem and two 2-Dproblems. The related spectral schemes are proposed. The convergence is proved. In particular, bychoosing appropriate basis functions with zero slope at the endpoint, a n-diagonal stiffness matrixis employed, instead of the full stiffness matrices encountered in the classical spectral method. Wealso present some numerical results to demonstrate the efficiency of this approach.