Odd behaviour of even geometries: an explanation for superconvergent geometric consistency errors

Piecewise polynomial surface approximations used in surface finite element methods often seem to behave better than their standard approximation properties suggest if their polynomial order is even. We explain this superconvergence through cancellation of leading interpolation errors on suitably structured meshes that naturally arise in some refinement processes. This cancellation improves weighted integral estimates for functions, derivatives, and geometric quantities. Applications include estimates for surface normals, the Weingarten map, and Gaussian curvature. Numerical experiments reproduce the predicted parity-dependent behaviour and support the proposed explanation of superconvergent geometric consistency errors, while the corresponding pointwise errors retain their standard orders.