We present a novel methodology for deriving high-order volume elements (HOVE) designed for the integration of scalar functions over regular embedded manifolds. For constructing HOVE we introduce square-squeezing --a homeomorphic multilinear hypercube-simplex transformation reparametrizing an initial flat triangulation of the manifold to a cubical 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，用于推导设计用于在正则嵌入流形上积分标量函数的高阶体积元（HOVE）。为构建HOVE，我们引入了平方挤压——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新参数化为立方体网格。通过采用平方挤压，我们借助切比雪夫-洛巴托网格上的插值，对重参数化网格的每个超立方体区域上的被积函数和体积元进行逼近。该策略规避了龙格现象，将初始积分替换为可通过高阶求积精确计算的闭式表达式。我们证明了关于被积函数和曲面参数化的r阶全变差的新颖积分误差界，预测了仅随插值次数缩放的高代数逼近率，而非如常见情况那样依赖于平均单纯形尺寸。对于全变差随r增加而恒定有界的光滑积分，估计证明积分误差甚至呈指数下降，而网格细化仅能达到代数收敛率。所得逼近能力在多个数值实验中得以验证，特别展示了p细化在克服h细化对高度变化光滑积分的局限性方面的优势。