We consider linear second order differential equation y''= f with zero Dirichlet boundary conditions. At the continuous level this problem is solvable using the Green function, and this technique has a counterpart on the discrete level. The discrete solution is represented via an application of a matrix -- the Green matrix -- to the discretised right-hand side, and we propose an algorithm for fast construction of the Green matrix. In particular, we discretise the original problem using the spectral collocation method based on the Chebyshev--Gauss--Lobatto points, and using the discrete cosine transformation we show that the corresponding Green matrix is fast to construct even for large number of collocation points/high polynomial degree. Furthermore, we show that the action of the discrete solution operator (Green matrix) to the corresponding right-hand side can be implemented in a matrix-free fashion.

翻译：我们研究具有零狄利克雷边界条件的线性二阶微分方程 y'' = f。在连续层面，该问题可通过格林函数求解，此技术在离散层面亦有对应方法。离散解通过矩阵（格林矩阵）作用于离散化右侧项表示，本文提出一种快速构造格林矩阵的算法。具体而言，我们采用基于切比雪夫-高斯-洛巴托点的谱配置法离散原问题，并借助离散余弦变换证明：即使配置点数量巨大/多项式次数较高，对应格林矩阵仍可快速构造。此外，我们证明离散解算子（格林矩阵）对相应右侧项的作用可实现无矩阵化计算。