Before revision:  We show our methods have high-order convergence rates for the metric and other geometric quantities of the manifold , for the truncation errors of exterior calculus operators , and for the solution errors of the Stokes problem for hydrodynamic flows on curved surfaces. Our approaches also may be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.
After revision:  We show our methods have high-order convergence rates for the metric and other geometric quantities of the manifold , for the truncation errors of exterior calculus operators . The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.