Consistent approximations of several geometric differential operators and their convergence

The numerical integration of geometric partial differential equations is used in many applications such as image processing, surface processing, computer graphics and computer-aided geometric design. Discrete approximations of several first- and second-order geometric differential operators, such as the tangential gradient operator, the second tangential operator, the Laplace–Beltrami operator and the Giaquinta–Hildebrandt operator, are utilized in the numerical integrations. In this paper, we consider consistent discretized approximations of these operators based on a quadratic fitting scheme. An asymptotic error analysis is conducted which shows that under very mild conditions the discrete approximations of the first- and second-order geometric differential operators have quadratic and linear convergence rates, respectively.