Title: Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator Show Chinese title

Title: 使用拉普拉斯-贝尔特拉米算子进行隐式定义体积的高精度数值计算

Abstract: This paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso-contour of a sufficiently smooth level-set function. After spatial discretization, local approximation of the hypersurface and application of the Gaussian divergence theorem, the volume integrals are transformed to surface integrals. Application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal domains, showing both high accuracy and thrid- to fourth-order convergence in space. Show Chinese abstract

Abstract: 本文介绍了一种新颖的方法，用于高效且准确地计算由可定向超曲面作为边界的一个区域的体积，该超曲面作为足够平滑的水平集函数的等值线隐式给出。 在空间离散化后，对超曲面进行局部近似并应用高斯散度定理，体积积分被转换为面积分。 应用面积分散度定理可以进一步减少为线积分，这对于数值求积是有利的。 我们讨论了理论基础并提供了数值算法的详细信息。 最后，我们给出了嵌入在长方体域中的凸和非凸超曲面的数值结果，显示了在空间中具有高精度和三到四阶收敛性。