Highly Accurate Rational-Spectral Method And Petrov-Galerkin Algorithm

Function approximation is a classical topic in scientific computation.Polynomial approximation plays a significant role in function approximation,numerical analysis and algorithmic implementation.Convergence rate of the polynomial approximation can be characterised by the Bernstein theorem for analytic functions.While for singular function,the uniform convergence rate of the polynomial approximation is well established,which indicates an algebraically convergent rate.Besides polynomial case,rational polynomials have been widely concerned for its superior in approximating singular functions.In this thesis,with the aid of the scaled map,we are going to establish two kinds of rational approximations based on the barycentric rational formulation of the second kind to singular functions at the origin.For the Volterra integral equation of the second kind with the singular kernel,we propose a kind of numerical scheme built upon the series solution near the origin and the conformal map.Our theoretical analysis is agreed well with the numerical experiments.In addition,for the spectral-Galerkin scheme for the initial problems,with appropriate choice of the basis functions such that the stiffness matrix is identity matrix and the mass matrix is diagonally(but non-symmetrical)sparse,we characterise the distributions of the eigenvalues of the mass matrices,which can be utilized in algorithm implementation.The main contributions of this thesis are as follows:(1)For functions with singularity at the origin,by the scaled map and the barycentric formula of the second kind,we establish two kinds of rational interpolation schemes.One scheme is based on the scaled Chebyshev points of the second kind and the simplified barycentric weights of the Cheybshev points of the second kind,and the other is proposed at the scaled equispaced nodes and the simplified weighs of the Floater-Hormann rational formula at the equispaced nodes.Some properties of the proposed schemes are analysed and the convergence rates are illustrated through ample numerical experiments.Comparing with the existed algorithms,we demonstrate the superiors of the rational formulae.Some applications are considered which illustrate the efficiency and robustness of the proposed rational schemes.(2)As the solution of the Volterra integral equation of the second kind with the singular kernel is singular near the left boundary point,some properties of the solution can be analysed with the aid of the Picard iteration.The solution can be written in series form near the singular point.By splitting the integral interval into two parts,we propose an efficient numerical scheme by applying the series solution in the first small interval and appealing to the conformal map at the second interval.Remarkably,the proposed scheme is independent of the choice of the samples in the second interval.Theoretical analysis implies that the proposed scheme possesses exponential convergence rate,which is illustrated by numerical experiments.(3)For the initial value problems,we study the eigenvalues of the mass matrix of the spectral Galerkin schemes.With appropriate choice of the basis functions,the stiffness matrix reduces to the identity matrix,and the mass matrix is diagonally sparse.For the first-order initial value problem,we prove the equivalence between the collocation method at the Gauss points and the Petrov-Galerkin method and demonstrate the relation between the eigenvalues of the mass matrix and the zeros of the Bessel polynomials.Furthermore,for the dual Petrov-Galerkin scheme,we can show that the eigenvalues of the mass matrix of the dual Petrov-Galerkin method are related to the zeros of the generalised Bessel polynomials.For the second-order initial value problems,the eigenvalues of the mass matrix are related to the square of the zeros of one kind of the generalised Bessel polynomials.Generally,for the m-th order initial value problems,we can also characterise the corresponding eigenvalues of the mass matrix to the zeros of one kind of the generalised Bessel polynomials.Then our analysis on the eigenvalues can be applied to the algorithms in solving the evolutionary problems,such as QZ algorithms and diagonalisation algorithms.Pros and cons of these two algorithms are illustrated with various numerical experiments.