Superconvergence of linear finite elements on simplicial meshes

• Main Author: Chen, Jie
• Other Authors: Wang Desheng
• Format: Theses and Dissertations
• Language: English
• Published: 2012
• Subjects: DRNTU::Science::Mathematics
• Online Access: https://hdl.handle.net/10356/50615
• Institution: Nanyang Technological University

In this dissertation work, we present a theoretical analysis for linear finite element gradient superconvergence on three-dimensional simplicial meshes where the lengthes of each pair of opposite edges in most tetrahedrons differ only by $O(h^{1+\alpha})$. We first derive a local error expansion formula in $n$ dimensional spaces and then use this identity to analyze the interpolantwise gradient superconvergence on simplicial meshes. In three dimensional spaces, we show that the gradient of the linear finite element solution $u_h$ is superconvergent to the gradient of the linear interpolation $u_I$ with an order $O(h^{1+\rho})(0<\rho\leq \alpha)$. Numerical examples are presented to verify the theoretical result. In four dimensional spaces, we find that there is no simplicial mesh that satisfies the edge pair condition.