Superconvergent quadriatic finite element on uniform tetrahedral meshes

TL;DR

This paper demonstrates superconvergence properties of quadratic finite elements on uniform tetrahedral meshes, showing improved convergence and a method to enhance solutions to cubic accuracy, supported by numerical validation.

Contribution

The paper proves superconvergence of quadratic finite elements on uniform tetrahedral meshes and introduces a technique to elevate solutions to cubic accuracy.

Findings

Proves $H^1$ and $L^2$ convergence of $P_2$ finite elements.

Establishes superconvergence of $P_2$ solutions on uniform tetrahedral meshes.

Provides a method to lift quadratic solutions to cubic accuracy.

Abstract

By a direct computation, we show that the $P_2$ interpolation of a $P_3$ function is also a local $H^1$-projection on uniform tetrahedral meshes, i.e., the difference is $H^1$-orthogonal to the $P_2$ Lagrange basis function on the support patch of tetrahedra of the basis function. Consequently, we show the $H^1$ and $L^2$ rconvergence of the $P_2$ Lagrange finite element on uniform tetrahedral meshes. Using the standard 20-points Lagrange $P_3$ interpolation, where the 20 nodes are exactly some $P_2$ global basis nodes, we lift the superconvergent $P_2$ finite element solution to a quasi-optimal $P_3$ solution on each cube. Numerical results confirm the theory.