The Legendre Tau Method For Nonlinear Partial Differential Equations And Its Domain Decomposition Method

The aim of this paper is to study the Legendre tau method for nonlinear partial differential equations.The tau method is one of the basic forms of spectral methods and has been used in many applications of numerical solutions of differential equations. Further-more, it seems that tau and, more generally, Petrov-Galerkin methods are more suitable for odd-order equations than Galerkin methods and collocation methods. However, error estimates of the tau method for even-order equations are sometimes not desirable and there are some comments that the tau method is inferior to the Galerkin and collocation methods in accuracy. These motivate our interest in the numerical analysis of the tau method.One of work in this thesis is that we prove the Legendre tau method has the optimal rate of convergence in L2-norm, H1-norm and H2-norm for one-dimensional second-order steady differential equations with three kinds of boundary conditions and in L2-norm for the corresponding evolution equation with the Dirichlet boundary condition. For the generalized Burgers equation, a Legendre tau-Chebyshev colloca-tion method is developed. For spatial discretization, we adopt so-called Chebyshev-Legendre method. For temporal discretization, the leapfrog/Crank-Nicolson scheme is used, which treats the linear terms implicitly and the nonlinear terms explic-itly. Optimal convergent in L2-norm can also be obtained. Finally, some numerical examples are given.Next, the Legendre tau method is considered to the two-dimensional case. The optimal convergence rate in H1-norm for two-dimensional Poisson equation is gotten. Through the duality method, the optimal convergence rate in L2-norm is also obtained. Then, the Legendre tau method for the vorticity equation is discussed. The stability and convergence of the semi-discrete scheme are given.Through careful analysis and numerical experiments, it is noted that the con-vergence of the tau method depends on the singularity of solutions. A Legendre Petrov-Galekin method for fourth-order differential equations is developed. The way to treat test functions of the Legendre Petrov-Galekin method is similar to that of the tau method, namely, that the test functions of the Leg-endre Petrov-Galekin method remove two boundary conditions. The optimal rate of convergence in L2-norm is gotten. A Legendre Petrov-Galerkin and Chebyshev collocation method is developed for the nonlinear Kuramoto-Sivashinsky equation. The work gives the optimal rate of convergence in L2-norm. The numerical experi-ments are given which demonstrate the efficient of proposed schemes.Finally, the Legendre tau methods to multidomain and first-order evolution equation with variable coefficients are discussed. The appropriate schemes are pre-sented, and the convergent results are also given.