Linear finite element superconvergence on simplicial meshes

Abstract

We study the linear finite element gradient superconvergence on special simplicial meshes which satisfy an edge pair condition. This special geometric condition implies that for most simplexes in the mesh, the lengths of each pair of opposite edges in each 3-face are assumed to differ only by $O(h^{1+\alpha })$ for some constant $\alpha >0$, with $h$ being the mesh parameter. To analyze the interplant gradient superconvergence, we present a local error expansion formula in general $n$ dimensional space which also motivates the condition on meshes. In the three dimensional space, we show that the gradient of the linear finite element solution $u_h$ is superconvergent to the gradient of the linear interpolatant $u_I$ with an order $O(h^{1+\rho })$ for $0<\rho \leq \alpha$. Numerical examples are presented to verify the theoretical findings. While we illustrate that tetrahedral meshes satisfying the edge pair condition can often be produced in three dimension, we also show that this may not be the case in higher dimensional spaces.