标题： 二阶常微分方程离散格林算子的快速构造 显示英文标题

标题： Fast construction of the discrete Green operator for a second order ordinary differential equation

摘要： 我们考虑带有零狄利克雷边界条件的二阶线性微分方程 y''= f。 在连续层面，这个问题可以通过格林函数来求解，这种技术在离散层面上也有对应的实现。 离散解通过将离散化的右边应用一个矩阵——格林矩阵——来表示，我们提出了一种快速构建格林矩阵的算法。 特别是，我们使用基于切比雪夫-高斯-洛巴托点的谱配置方法对原问题进行离散化，并利用离散余弦变换，我们证明即使对于大量配置点/高多项式次数，相应的格林矩阵也可以快速构建。 此外，我们表明离散求解算子（格林矩阵）作用于相应右边的操作可以以无矩阵的方式实现。 显示英文摘要

摘要： 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.