This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The sparse grid method is a popular technique for high dimensional problems, and the associated collocation method has been well studied in the literature. The contribution of this work is the introduction of a systematic framework for collocation onto high-order piecewise polynomial space that is allowed to be discontinuous. We consider both Lagrange and Hermite interpolation methods on nested collocation points. Our construction includes a wide range of function space, including those used in sparse grid continuous finite element method. Error estimates are provided, and the numerical results in function interpolation, integration and some benchmark problems in uncertainty quantification are used to compare different collocation schemes.

将自适应稀疏网格配置方法构造到任意阶分段多项式空间上。稀疏网格法是解决高维问题的一种流行技术，相关的配置方法已经在文献中得到了很好的研究。这项工作的贡献是引入了一个系统框架，用于配置到允许不连续的高阶分段多项式空间上。我们在嵌套搭配点上同时考虑了Lagrange和Hermite插值方法。我们的构造包括广泛的功能空间，包括稀疏网格连续有限元方法中使用的功能空间。提供误差估计，并使用函数插值，积分和不确定性量化中的一些基准问题的数值结果来比较不同的配置方案。