Superconvergence and Gradient Recovery of Linear Finite Elements for the Laplace–Beltrami Operator on General Surfaces

Abstract

Superconvergence results and several gradient recovery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the Laplace-Beltrami equation on general surfaces with mildly structured triangular meshes. For a large class of practically useful grids, the surface linear finite element solution is proven to be superclose to an interpolant of the exact solution of the Laplace-Beltrami equation, and as a result various postprocessing gradient recovery, including simple and weighted averaging, local and global $L^2$-projections, and Zienkiewicz and Zhu (Z-Z) schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution. Numerical experiments are presented to confirm the theoretical results.