Fully Diagonalized Legendre Spectral Methods

In this thesis,we study the diagonalized Legendre spectral methods using Sobolev orthogonal polynomials for elliptic boundary value problems.The paper includes two parts of research contents.Firstly,we study the fully diagonalized Legendre spectral methods using Sobolev orthogonal or biorthogonal polynomials for second order Neumann boundary value problems;Secondly,we study the fully diagonalized Legendre spectral methods using Sobolev orthogonal polynomials for second order Dirichlet boundary value problems.The main work of this thesis is divided into four chapters.In the first chapter,the significance and research status of spectral method are introduced,and the research results related to the content of this thesis are briefly introduced.The second chapter introduces some basic definitions and related properties of the Legendre spectral method,and gives some basic theorems which is closely related to the research contents in this thesis.In the third chapter,we construct Sobolev orthogonal or biorthogonal Legendre basic functions for two kinds of elliptic equations with Neumann boundary conditions,and study the fully diagonalized Legendre spectral methods.Numerical examples are presented to verify the effectiveness of the method.In the fourth chapter,we first construct Sobolev orthogonal Legendre basis functions for one-dimentional elliptic equation with Dirichlet boundary condition,and study the fully diagonalized Legendre spectral methods.Then,we extend this idea to the two-dimensional Dirichlet boundary value problems.Finally,we give some numerical results to exhibit the effectiveness and accuracy.