AI 中文总结
本文提出一种稳定化Morley有限元方法求解流函数形式的闭合曲面Stokes问题，通过新几何估计获得最优收敛性，数值实验验证了理论结果。
AI 中文摘要
我们针对闭合曲面上流函数形式的曲面Stokes问题，提出了一种内在的稳定化Morley有限元方法。该方法直接在曲面的多面体近似上构建。引入了无参数的跳稳定项以恢复强制性，并建立了无迹Hessian的离散Korn不等式。主要的分析贡献是针对分段线性近似闭合曲面的法向分离几何估计，该估计表明，一类广泛的依赖法向的几何相容性误差实际上是二阶的，尽管直接处理仅能给出一阶控制。缺失的阶数通过一阶法向变分中的积分抵消来恢复。该估计改进了标准的$P_h\nu$型估计，并得到无迹Hessian、Stokes型张量格林恒等式以及离散化中出现的余法向通量的二阶相容性。结合离散Korn不等式，在自然$H^3$正则性下，这些估计给出了最优阶收敛：在 broken $H^2$范数下为一阶，在 broken $H^1$范数下为二阶。作为直接结果，恢复的切向速度具有二阶$L^2$估计。提供了数值实验以支持理论结果。
英文摘要
We propose an intrinsic stabilized Morley finite element method for the stream-function formulation of the surface Stokes problem on closed surfaces. The method is posed directly on a polyhedral approximation of the surface. A parameter-free jump stabilization is introduced to recover coercivity, and a discrete Korn's inequality for the trace-free Hessian is established. The main analytical contribution is a normal-separated geometric estimate for piecewise linearly approximated closed surfaces. It shows that a broad class of normal-dependent geometric consistency errors is in fact second order, even though a direct treatment suggests only first-order control. The missing order is recovered through an integral cancellation in the first normal variation. This estimate refines the standard $P_hν$-type estimate and yields second-order consistency for the trace-free Hessian, the Stokes-type tensor Green identity, and the conormal fluxes arising in the discretization. Together with the discrete Korn's inequality, these estimates yield, under the natural $H^3$ regularity, optimal-order convergence: first order in the broken $H^2$ norm and second order in the broken $H^1$ norm. As a direct consequence, the recovered tangential velocity admits a second-order $L^2$ estimate. Numerical experiments are provided to support the theoretical results.
