This work develops optimal preconditioners for the discrete H(curl) and H(div) problems on two-dimensional surfaces by nodal auxiliary space preconditioning (Hiptmair and Xu in SIAM J Numer Anal 45:2483–2509, 2007). In particular, on unstructured triangulated surfaces, we develop fast and user-friendly preconditioners for the edge and face element discretizations of curl–curl and grad–div problems based on inverting several discrete surface Laplacians. The proposed preconditioners lead to efficient iterative methods for computing harmonic tangential vector fields on discrete surfaces. Numerical experiments on two- and three-dimensional hypersurfaces are presented to test the performance of those surface preconditioners.

这项工作通过节点辅助空间预处理为二维表面上的离散H (curl) 和H (div) 问题开发了最优预条件子（Hiptmair 和 Xu in SIAM J Numer Anal 45:2483–2509, 2007）。特别是，在非结构化三角曲面上，我们开发了快速且用户友好的预处理器，用于基于反转多个离散曲面拉普拉斯算子的 curl-curl 和 grad-div 问题的边缘和面元素离散化。所提出的预条件子导致用于计算离散表面上的谐波切向矢量场的有效迭代方法。提出了二维和三维超曲面的数值实验以测试这些表面预调节器的性能。