Some progress in spectral methods Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday  

Some progress in spectral methods

作　　者：GUO BenYu 

机构地区：[1]Department of Mathematics, Shanghai Normal University [2]Scientific Computing Key Laboratory of Shanghai Universities [3]Division of Computational Science,E-institute of Shanghai Universities

出　　处：《Science China Mathematics》2013年第12期2411-2438,共28页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11171227);Fund for Doctoral Authority of China(Grant No.20123127110001);Fund for E-institute of Shanghai Universities(Grant No.E03004);Leading Academic Discipline Project of Shanghai Municipal Education Commission(Grant No.J50101)

摘　　要：In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applica- tions to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spec- tral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.In this paper,we review some results on the spectral methods.We frst consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems,including degenerated and singular diferential equations.Then we present the generalized Jacobi quasi-orthogonal approximation and its applications to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions.We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains.Next,we consider the Hermite spectral method and the generalized Hermite spectral method with their applications.Finally,we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defned on unbounded domains.We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.

关 键 词：JACOBI Hermite and Laguerre spectral approximations Jacobi and Laguerre quasi-orthogonalapproximations spectral and spectral element methods degenerated and singular problems problems on non-rectangular and unbounded domains problems of non-standard type exterior problems 

分 类 号：O174[理学—数学]