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

Authors:
Huayi Wei;Long Chen;Yunqing Huang
Affiliations:
huayiwei1984@gmail.com and huangyq@xtu.edu.cn;chenlong@math.uci.edu;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2010

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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.