Stabilized Morley FEM for surface Stokes in stream-function formulation: Optimal convergence via a new geometric estimate

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 $P_h\nu$-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.