In this article we consider the application of Euler's homogeneous function theorem together with Stokes' theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two- and three-dimensions, respectively. This approach allows for the integrals to be evaluated based on only computing the values of the integrand and its derivatives at the vertices of the polytopic domain, without the need to construct a sub-tessellation of the underlying domain of interest. Here, we present a detailed analysis of the computational complexity of the proposed algorithm and show that this depends on three key factors: the ambient dimension of the underlying polytopic domain; the size of the requested polynomial space to be integrated; and the size of a directed graph related to the polytopic domain. This general approach is then employed to compute the volume integrals arising within the discontinuous Galerkin finite element approximation of the linear transport equation. Numerical experiments are presented which highlight the efficiency of the proposed algorithm when compared to standard quadrature approaches defined on a sub-tessellation of the polytopic elements.

翻译：本文考虑应用欧拉齐次函数定理结合斯托克斯定理，在二维和三维空间中分别对一般多边形和多面体（多胞体）域上的多项式空间族进行精确积分。该方法仅需计算被积函数及其在多胞体域顶点处的值及导数即可评估积分，无需构建目标域的子剖分。我们详细分析了所提算法的计算复杂度，表明其取决于三个关键因素：多胞体域的覆盖维度、待积分多项式空间的大小，以及与该多胞体域相关的有向图尺寸。随后将该通用方法应用于线性输运方程的间断伽辽金有限元近似中涉及的体积积分计算。数值实验展示了所提算法相较于传统基于多胞体单元子剖分的求积方法的高效性。