Superconvergent quadriatic finite element on uniform tetrahedral meshes

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.