In this paper, we propose a sparse spectral-Galerkin approximation scheme for solving the second-order partial differential equations on an arbitrary tetrahedron. Generalized Koornwinder polynomials are introduced on the reference tetrahedron as basis functions with their various recurrence relations and differentiation properties being explored. The method leads to well-conditioned and sparse linear systems whose entries can either be calculated directly by the orthogonality of the generalized Koornwinder polynomials for differential equations with constant coefficients or be evaluated efficiently via our recurrence algorithm for problems with variable coefficients. Clenshaw algorithms for the evaluation of any polynomial in an expansion of the generalized Koornwinder basis are also designed to boost the efficiency of the method. Finally, numerical experiments are carried out to illustrate the effectiveness of the proposed Koornwinder spectral method.

翻译：在本文中,我们提出一个稀疏的光谱-伽勒金近似方案,以解决对任意四面体的二阶部分偏差方程式。在参考四面体中引入了通用的Koornwinder多面体,作为四面体的各种复发关系和差异特性的基础功能。该方法导致条件良好和分散的线性系统,其条目可以直接根据通用的Koornwinder多面体对等方方方对不变系数的正对数计算,或者通过我们针对可变系数问题的复现算法加以有效评估。在扩大通用的Koorn风器基础时,用于评价任何多面体的Clenshaw算法也旨在提高该方法的效率。最后,进行了数字实验,以说明拟议的Koorn风器光谱法的有效性。