Abstract In this paper, we develop and analyze a new divergence-free kernel approximation method for the time-dependent incompressible Stokes equations on surfaces. The novelty of our proposed method comes from the surface Helmholtz decomposition, which can convert the surface Stokes equations into a coupled equations in which the velocity is deadly divergence-free so that no inf–sup conditions have to be satisfied. Spatial discretization of velocity is implemented by the radial kernel divergence-free approximation spaces, which only need scatter nodes on surfaces. Using the radial basis function collocation method in space, we derive the rigorous stability and convergence result. Numerical examples are presented, demonstrating the efficiency of some model problems on more general surfaces.

中文翻译：

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基于表面 Helmholtz 分解的表面 Stokes 方程无发散径向核

摘要 在本文中，我们开发并分析了一种新的无散度核近似方法，用于曲面上的瞬态不可压缩 Stokes 方程。我们提出的方法的新颖性来自表面 Helmholtz 分解，它可以将表面 Stokes 方程转换为耦合方程，其中速度是致命的无发散，因此不必满足 inf-sup 条件。速度的空间离散化是通过径向核无散度近似空间实现的，它只需要表面上的散射节点。使用空间中的径向基函数搭配方法，我们得到了严格的稳定性和收敛性结果。给出了数值例子，证明了一些模型问题在更一般的表面上的效率。