Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases  

作　　者：Xiaolong Zhang John P.Boyd 

机构地区：[1]MOE-LCSM,School of Mathematics and Statistics,Hunan Normal University,Changsha 410081,China [2]Department of Climate&Space Sciences and Engineering,University of Michigan,Ann Arbor,MI 48109,USA

出　　处：《Science China Mathematics》2023年第1期191-220,共30页中国科学：数学（英文版）

基　　金：supported by National Science Foundation of USA (Grant No. DMS1521158);National Natural Science Foundation of China (Grant No. 12101229);the Hunan Provincial Natural Science Foundation of China (Grant No. 2021JJ40331);the Chinese Scholarship Council (Grant Nos. 201606060017 and 202106720024)。

摘　　要：When one solves differential equations by a spectral method,it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients anto modified basis functions that incorporate the boundary conditions.For homogeneous Dirichlet boundary conditions,u(±1)=0,popular choices include the "Chebyshev difference basis" ζn(x)≡Tn+2(x)-Tn(x) with coefficients here denoted by bnand the "quadratic factor basis" Qn(x)≡(1-x2)Tn(x) with coefficients cn.If u(x) is weakly singular at the boundary,then the coefficients andecrease proportionally to O(A(n)/nκ) for some positive constant κ,where A(n) is a logarithm or a constant.We prove that the Chebyshev difference coefficients bndecrease more slowly by a factor of 1/n while the quadratic factor coefficients cndecrease more slowly still as O(A(n)/nκ-2).The error for the unconstrained Chebyshev series,truncated at degree n=N,is O(|A(N)|/Nκ) in the interior,but is worse by one power of N in narrow boundary layers near each of the endpoints.Despite having nearly identical error norms in interpolation,the error in the Chebyshev basis is concentrated in boundary layers near both endpoints,whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x.Meanwhile,for Chebyshev polynomials,the values of their derivatives at the endpoints are O(n2),but only O(n) for the difference basis.Furthermore,we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases,solved by the least squares method.We also find an interesting fact that on the face of it,the aliasing error is regarded as a bad thing;actually,the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation.But the premise is under the same basis,and when involving different bases,it may not be established yet.

关 键 词：Chebyshev polynomial interpolation endpoint singularities least squares method 

分 类 号：O174[理学—数学]