We propose a Petrov--Galerkin spectral method for ODEs with variable coefficients. When the variable coefficients are smooth, the new method yields a strictly banded linear system, which can be efficiently constructed and solved in linear complexity. The performance advantage of our method is demonstrated through benchmarking against Mortensen's Galerkin method and the ultraspherical spectral method. Furthermore, we introduce a systematic approach for designing the recombined basis and establish that our new method serves as a unifying framework that encompasses all existing banded Galerkin spectral methods. This significantly addresses the ongoing challenge of developing recombined bases and sparse Galerkin spectral method. Additionally, the accelerating techniques presented in this paper can also enhance the performance of the ultraspherical spectral method.

翻译：本文提出了一种针对变系数常微分方程的Petrov-Galerkin谱方法。当变系数函数光滑时，新方法将生成严格带状的线性系统，该系统可在线性复杂度下高效构建与求解。通过与Mortensen的Galerkin方法及超球谱方法进行基准测试，验证了本方法的性能优势。此外，我们提出了一种设计重组基函数的系统化途径，并证明新方法可作为统一框架涵盖所有现有带状Galerkin谱方法。这显著解决了当前重组基构造与稀疏Galerkin谱方法发展中的核心难题。同时，本文提出的加速技术亦可提升超球谱方法的计算性能。