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

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 LaplaceBeltrami 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 post-processing gradient recovery, including simple and weighted averaging, local and global L2-projections, and 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.