AI 中文总结
研究比较在样条曲线跟踪边界的平面单元上构建代数求积规则的四种方法，基于格林定理等理论，讨论优缺点并提供开源Matlab代码，为曲线多面体单元方法应用提供参考。
AI 中文摘要
我们比较了在平面单元上构建代数求积规则的四种方法，这些平面单元的边界由样条跟踪。我们在过去二十年中开发的这些方法基于格林定理以及多项式逼近理论的一些基石：高斯求积、恰卡洛夫定理、离散切比雪夫展开（超插值）、类费克特点插值。我们从应用于曲线多面体单元方法的角度讨论了它们的优缺点。我们还在一个网站上免费提供了相应的开源Matlab代码。
英文摘要
We compare four methods for the construction of algebraic cubature rules on planar elements, whose boundary is tracked by splines. The methods, that we have developed over the last two decades, are based on Green theorem together with some cornerstones of polynomial approximation theory: Gaussian quadrature,Tchakaloff theorem, discretized Chebyshev expansion (hyperinterpolation), Fekete-like interpolation. We discuss their advantages and drawbacks in view of the application to curved polytopal element methods. We have also made freely available at a single site the corresponding open-source Matlab codes.
