Discrete schemes for Gaussian curvature and their convergence

The popular angular defect schemes for Gaussian curvature only converge at the regular vertex with valence 6. In this paper, we present a new discrete scheme for Gaussian curvature, which converges at the regular vertex with valence greater than 4. We show that it is impossible to build a discrete scheme for Gaussian curvature which converges at the regular vertex with valence 4 by a counterexample. We also study the convergence property of other discrete schemes for Gaussian curvature and compare their asymptotic errors by numerical experiments.