We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

中文翻译：

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曲面上流体动力流流形的无网格方法：广义移动最小二乘法 (GMLS) 方法

我们利用广义移动最小二乘法 (GMLS) 来开发用于离散流形上的流体动力流动问题的无网格技术。我们使用外部微积分来制定 Stokesian 体系中的不可压缩流体动力学方程，并通过广义矢量势处理无发散约束。这提供了较少以坐标为中心的描述，并且能够根据二阶椭圆算子系统开发四阶控制方程的有效数值方法和分裂方案。使用霍奇分解，我们开发了具有球形拓扑结构的流形的方法。我们展示了这些方法在求解曲面上的流体动力学流动时表现出高阶收敛率。这些方法还提供了度量、曲率、以及流形和相关外部微积分算子的其他几何量。这些方法还可用于为流形上的其他标量值和向量值问题开发高阶求解器。