In this work, we analytically identify natural superconvergent points of function values and gradients for triangular elements. Both the Poisson equation and the Laplace equation are discussed for polynomial finite element spaces (with degrees up to 8) under four different mesh patterns. Our results verify computer findings of , especially, we confirm that the computed data have 9 digits of accuracy with an exception of one pair (which has 8-7 digits of accuracy). In addition, we demonstrate that the function value superconvergent points predicted by the symmetry theory  are the only superconvergent points for the Poisson equation. Finally, we provide function value superconvergent points for the Laplace equation, which are not reported elsewhere in the literature.
Number in Series
Applied Mathematics | Mathematics | Numerical Analysis and Computation
AMS Subject Classification
Zhang, Zhimin and Lin, Runchang, "Natural Superconvergent Points of Triangular Finite Elements" (2003). Mathematics Research Reports. 12.