We present a high-order method that provides numerical integration on volumes, surfaces, and lines defined implicitly by two smooth intersecting level sets. To approximate the integrals, the method maps quadrature rules defined on hypercubes to the curved domains of the integrals. This enables the numerical integration of a wide range of integrands since integration on hypercubes is a well known problem. The mappings are constructed by treating the isocontours of the level sets as graphs of height functions. Numerical experiments with smooth integrands indicate a high-order of convergence for transformed Gauss quadrature rules on domains defined by polynomial, rational, and trigonometric level sets. We show that the approach we have used can be combined readily with adaptive quadrature methods. Moreover, we apply the approach to numerically integrate on difficult geometries without requiring a low-order fallback method.

翻译：我们提出了一种高阶方法，用于在由两个光滑相交水平集隐式定义的体积、曲面和线上进行数值积分。该方法通过将定义在超立方体上的求积规则映射到积分曲面域上来逼近积分，从而使得数值积分能够处理广泛类型的被积函数，因为超立方体上的积分是一个已知问题。映射的构建基于将水平集的等值线视为高度函数图形。针对由多项式、有理函数和三角函数水平集定义的域，对光滑被积函数进行的数值实验表明，经过变换的高斯求积规则具有高阶收敛性。我们展示了该方法可便捷地与自适应求积方法结合使用。此外，该方法可应用于困难几何形状的数值积分，无需依赖低阶备用方法。