Meshfree discretization of surface partial differential equations is appealing, due to their ability to naturally adapt to deforming motion of the underlying manifold. In this work, we consider an existing scheme proposed by Liang et al. reinterpreted in the context of generalized moving least squares (GMLS), showing that existing numerical analysis from the GMLS literature applies to their scheme. With this interpretation, their approach may then be unified with recent work developing compatible meshfree discretizations for the div-grad problem in \(\mathbb {R}^d\). Informally, this is analogous to an extension of collocated finite differences to staggered finite difference methods, but in the manifold setting and with unstructured nodal data. In this way, we obtain a compatible meshfree discretization of elliptic problems on manifolds which is naturally stable for problems with material interfaces, without the need to introduce numerical dissipation or local enrichment near the interface. We provide convergence studies illustrating the high-order convergence and stability of the approach for manufactured solutions and for an adaptation of the classical five-strip benchmark to a cylindrical manifold.

中文翻译：

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兼容的表面PDE的无网格离散化

曲面偏微分方程的无网格离散化很有吸引力，因为它们具有自然适应底层歧管变形运动的能力。在这项工作中，我们考虑了Liang等人提出的现有方案。在广义移动最小二乘（GMLS）的背景下进行了重新解释，表明来自GMLS文献的现有数值分析适用于其方案。通过这种解释，他们的方法可以与最近的工作统一起来，这些工作针对\（\ mathbb {R} ^ d \）中的div-grad问题开发了兼容的无网格离散化。。非正式地，这类似于将并置的有限差分扩展为交错的有限差分方法，但是在流形设置中以及具有非结构化的节点数据。通过这种方式，我们获得了歧管上椭圆问题的兼容无网格离散化，对于材料界面问题自然稳定，而无需在界面附近引入数值耗散或局部富集。我们提供的收敛性研究说明了制造解决方案的方法的高阶收敛性和稳定性，以及经典的五线基准对圆柱歧管的适应性。