Chebyshev-Legendre Spectral Method And Its Domain Decomposition Method

In this thesis, we study the Chebyshev-Legendre spectral methods and their domain decomposition methods.First, we introduce the approximation properties of Chebyshev interpolation operator in one dimension with single domain and multi-domains in general L2-norm, and also in several dimensions with single domain and multi-domains. These properties are essential in numerical analysis of Chebyshev-Legendre methods.Then we analyze the Chebyshev-Legendre spectral method for the generalized Burgers equation. Besides the Chebyshev-Legendre collocation (CLC) method in [15], a Leg-endre Galerkin Chebyshev collocation (LGCC) scheme is presented. The LGCC scheme is basically formulated in the Legendre Galerkin form but with the nonlinear term being treated with the Chebyshev collocation method. Here the Chebyshev-Gauss-Lobatto (CGL) points are adopted. By combining the Galerkin and collocation methods, the scheme seems more flexible and easier to be generalized to multi-domain approaches. In numerical analysis of such methods, we need to consider the stability and approximation properties of the Chebyshev interpolation operator in L2-norm rather than in the Chebyshev weighted norm. Also, due to the property of the Chebyshev interpolation operator, we find that it is difficult to get the desired L2-estimate directly for our fully discrete scheme. This is why an H1-estimate is involved in analysis. Optimal convergence rate of the methods is obtained through combining L2-and H1-estimates.Next we apply a multi-domain Legendre Galerkin Chebyshev collocation ( MLGCC) method to the Burgers equation. That means the LGCC methods are applied on all the subintervals. We also introduce appropriate base functions as in [104] and [72] so that the matrix of the system is sparse and the problem in each subinterval can be solved efficiently and in parallel. The stability and the optimal rate of convergence of the method are proved. Numerical results are given for both single domain and multi-domain methods to make a comparison.After applying the Chebyshev-Legendre method to the one-dimensional problems, we also treat two-dimensional vorticity equations with LGCC methods. Similar to the Burgers equation, we give the discrete schemes for the vorticity equations. That is basicallyformulated in the Legendre Galerkin form but the nonlinear terms are interpolated by Chebyshev collocation methods at Chebyshev-Gauss points.We take proper base functions to make the coefficient matrix sparse and also describe the algorithm. Then we give the optimal error estimates and numerical examples.At last, we study the Chebyshev-Legendre methods for the two-dimensional Navier-Stokes equations. We first construct a weak form that the velocity and pressure are decoupled. We do this as following: Separate the Sobolev space H01( ) into two spaces(the divergence-free functions space Vdiv and its orthogonal complement space Vdiv which are orthogonal each other. By utilizing the separation of the space H01, we can get the weak form of Navier-Stokes equations that the velocity and pressure are decoupled. For the velocity functions are belong to the divergence-free functions space, we also can get the weak forms with a modified nonlinear terms. Then we construct the Chebyshev-Legendre spectral schemes at single domain and multi-domains from the modified weak forms. At the end we can get the optimal error estimates for the velocity of the method with the single domain and multi-domains. We also analyze the estimate for the pressure at single domain and multi-domains.