| Spectral methods for Dirichlet boundary value problems have been developed extensively inthe past three decades. However, it is also interesting and important to consider various problemswith Neumann boundary condition. In a standard variational formulation, this kind of boundarycondition is commonly imposed in a natural way. However, this approach usually leads to a fullstiffness matrix for approximating the second derivatives.In this dissertation, we investigate some Neumann problems, using the Jacobi spectral methodwith essential imposition of Neumann condition. This method differs from the classical spectralmethods for such problems, the homogeneous Neumann boundary condition is satisfied exactlyfor each basis. The main advantage of such treatment consists in that, a band stiffness matrix isemployed for one-dimensional problems, instead of the full stiffness matrix encountered in theclassical variational formulation.For analyzing the numerical errors, we establish some basic results on Jacobi quasi-orthogonalapproximations for Neumann problems. The convergence of proposed schemes is proved. We alsopresent some numerical results to demonstrate the efficiency of this approach.This work consists of three parts. In Chapter 1, we recall the history of numerical methods forNeumann problems. We also describe the motivation and the difficulties.In Chapter 2, we recall some basic properties of Jacobi polynomials. We establish somebasic results on the Jacobi quasi-orthogonal approximations. As examples, we propose the spec-tral schemes for one-dimensional Neumann problems. The convergence of proposed schemes isproved. Numerical results demonstrate the efficiency of this approach.In Chapter 3, we establish some basic results on the mixed Fourier-Jacobi approximation. Asexamples, we propose the mixed spectral schemes for two-dimensional Neumann problems. Theconvergence of proposed schemes is proved. Numerical results also show the efficiency of thisapproach.
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