表面Stokes问题的无压力虚拟元方法
A Pressure-Free Virtual Element Method for the Surface Stokes Problem
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中文总结 AI 辅助
针对任意亏格闭曲面上的表面Stokes问题，提出一种基于非协调Stokes复形的无压力虚拟元方法，通过顶点边约束实现切向连续性，无需惩罚项，并证明速度在破裂H^1和L^2范数下分别达到最优一阶和二阶收敛，数值实验验证了理论。
中文摘要 AI 辅助
我们针对任意亏格闭曲面的多边形逼近上的表面Stokes问题，发展了一种无压力虚拟元方法。该方法建立在具有交换插值和正确离散上同调的非协调Stokes复形之上。其速度空间是精确切向且H(div)协调的，而基于顶点的边约束在不增加额外自由度的前提下强制了切向边平均值的连续性。由此得到的格式无需惩罚项，并且局部的保散度重构提供了一个可计算的、压力鲁棒的载荷。我们还构造了该复形的一个显式宏单元实现，并证明了相应的诱导虚拟格式在代数上等价于一个直接的宏单元Galerkin方法。我们建立了稳定性，并在破裂H^1范数下获得了速度的最优一阶收敛性，在L^2范数下获得了二阶收敛性。边约束提供了恢复最优L^2收敛速率所需的二阶弱一致性估计。数值实验证实了理论结果。
英文摘要
We develop a pressure-free virtual element method for the surface Stokes problem on polygonal approximations of closed surfaces of arbitrary genus. The method is built on a nonconforming Stokes complex with commuting interpolation and the correct discrete cohomology. Its velocity space is exactly tangential and \(H(\operatorname{div})\)-conforming, while a vertex-based edge constraint enforces continuity of tangential edge averages without additional degrees of freedom. The resulting formulation requires no penalties, and a local divergence-preserving reconstruction provides a computable pressure-robust load. We also construct an explicit macroelement realization of the complex and show that the corresponding induced virtual formulation is algebraically equivalent to a direct macroelement Galerkin method. We establish stability together with optimal first-order convergence in the broken \(H^1\) norm and second-order convergence in the \(L^2\) norm for the velocity. The edge constraint yields the second-order weak consistency estimate needed to recover the optimal \(L^2\) rate. Numerical experiments confirm the theoretical results.
