The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory, the Laplace-Beltrami operator can be pre-/post-conditioned with an integral operator whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard~$1/r$ kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace-Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries.

翻译：嵌入三维空间中的封闭曲面上的拉普拉斯-贝尔特拉米问题出现在物理学的许多领域，包括分子动力学（表面扩散）、电磁学（调和向量场）和流体动力学（囊泡变形）。利用经典势理论，拉普拉斯-贝尔特拉米算子可以通过核函数具有平移不变性的积分算子进行预处理/后处理，从而得到条件良好的第二类弗雷德霍姆积分方程。这些方程具有势理论中的标准~$1/r$核函数，因此可以通过快速多极子方法（FMM）和高阶求积校正的组合快速且精确地求解。在本文中，我们详细描述了这样一种方案，提出了拉普拉斯-贝尔特拉米问题的两种替代积分公式，每种公式的解均可通过FMM加速获得。接着，我们展示了求解器的若干应用，重点聚焦于调和向量场的计算，这一计算与电磁学中的众多应用密切相关。针对每种应用，我们提供了一系列数值结果，详细展示了求解器在不同几何构型下的性能表现。