The Super-geometric Convergence Phenomenon Of Spectral Method For Differential Equations

There are mainly three numerical methods, which are finite difference method, finite element method and spectral method, for solving differential equations. Specially, spectral method is divided into three methods once again, which are Galerkin spectral methods, Tau method and collocation method. About spectral method, it has convergence of "infinite order". On the other word, if the solution of the original equation is infinite smooth, then the ap-proximate solution which is obtained by spectral method will converge to the exact solution with p-1 having arbitrary exponent, that is‖u - up‖≤Cp-α, The p corresponding to spectral method is generally far smaller than the p corresponding to finite difference or finite element method. Up to now, at home and aborad, there already have many studies on the error analysis of spectral method. For example, that Legendre collocation method having super-geometric convergence of the spectrum with the ultra-precision form of e-αp(logP-β)[31] was proved in theory.In this thesis, we focuses on observing the super-geometric convergence phenomenon from some examples, when applying the collocation method to solve differential equations. Firstly, we apply Fourier collocation method, Chebyshev collocation method and Legendre collocation method to solve some differential equations, e.g. second-order ordinary differential problem, one-dimensional and two-dimensional eigenvalue problem. Secondly, we analyze the error of the numerical solutions and its first derivative. By observing the figure about error, we find that above-methioned three methods which are used for numerical solution's approximation can reach supergeometric convergence, and the error estimates obtained in the problems also have the form of ultra-precision geometry of the spectrum.