Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if the solution to the partial differential equation is smooth enough. Naive implementations of kernel based methods suffer, however, from the cubic complexity in the degrees of freedom. Localized Lagrange bases have proven to overcome this computational complexity to some extend. In this article we present a rigorous proof for a geometric multigrid method with $\tau\ge 2$-cycle for elliptic partial differential equations on surfaces which is based on precomputed Lagrange basis functions. Our new multigrid provably works on quasi-uniform point clouds on the surface and hence does not require a grid-structure. Moreover, the computational cost scales log-linear in the degrees of freedom.

翻译：求解曲面上偏微分方程的核方法具有内在适应曲面几何的优势，若方程解足够光滑则可获得高逼近阶。然而，基于核方法的朴素实现受制于自由度三次方复杂度的限制。局部化拉格朗日基函数已被证明能在一定程度上克服这一计算复杂度。本文针对曲面上椭圆型偏微分方程，基于预计算拉格朗日基函数，提出了严格证明的$\tau\ge 2$循环几何多重网格法。该新型多重网格方法在曲面的准均匀点集上具有可证明的有效性，因此无需网格结构支持。此外，其计算复杂度随自由度呈对数线性增长。