We present a novel methodology for deriving high-order quadrature rules (HOSQ) designed for the integration of scalar functions over regular embedded manifolds. To construct the rules, we introduce square-squeezing--a homeomorphic multilinear hypercube-simplex transformation--reparametrizing an initial flat triangulation of the manifold to a hypercube mesh. By employing square-squeezing, we approximate the integrand and the volume element for each hypercube domain of the reparameterized mesh through interpolation in Chebyshev-Lobatto grids. This strategy circumvents the Runge phenomenon, replacing the initial integral with a closed-form expression that can be precisely computed by high-order quadratures. We prove novel bounds of the integration error in terms of the $r^\text{th}$-order total variation of the integrand and the surface parameterization, predicting high algebraic approximation rates that scale solely with the interpolation degree and not, as is common, with the average simplex size. For smooth integrals whose total variation is constantly bounded with increasing $r$, the estimates prove the integration error to decrease even exponentially, while mesh refinements are limited to achieve algebraic rates. The resulting approximation power is demonstrated in several numerical experiments, particularly showcasing $p$-refinements to overcome the limitations of $h$-refinements for highly varying smooth integrals.

翻译：我们提出了一种新颖的方法论，用于推导针对正则嵌入流形上标量函数的高阶求积规则（HOSQ）。为构造这些规则，我们引入了“方形挤压”——一种同胚的多线性超立方体-单纯形变换——将流形的初始平坦三角剖分重参数化为超立方体网格。通过采用方形挤压，我们利用切比雪夫-洛巴托网格上的插值来近似重参数化网格中每个超立方体域上的被积函数和体积元。该策略规避了龙格现象，将初始积分替换为一个可通过高阶求积精确计算的闭式表达式。我们证明了以被积函数和曲面参数化的$r$阶全变差为变量的积分误差新界限，预测出高阶代数逼近率仅与插值阶数相关，而非如通常情况那样依赖于平均单纯形尺寸。对于全变差随$r$增大而恒定有界的光滑积分，该估计证实积分误差甚至呈指数级下降，而网格细化只能达到代数级收敛速率。通过多个数值实验展示了所得逼近能力，特别针对高度振荡的光滑积分，展示了$p$细化如何克服$h$细化的局限性。