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高阶体积元（HOVE）构造方法，用于在规则嵌入流形上对标量函数进行积分。为构建HOVE，我们引入了方体压缩技术——一种同胚的多线性超立方体-单纯形变换，可将流形的初始平坦三角剖分重新参数化为立方体网格。通过采用方体压缩，我们在切比雪夫-洛巴托网格上通过插值来逼近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，将初始积分替换为可通过高阶求积公式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$r^\text{次}$阶全变差的新颖积分误差界，预测了仅随插值次数（而非通常的平均单纯形尺寸）缩放的高代数逼近率。对于全变差随$r$增大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，而网格细化仅能实现代数收敛率。最终获得的逼近能力在多个数值实验中得到验证，特别展示了$p$细化在克服$h$细化对高振荡光滑积分的局限性方面的优势。