3D $H^2$-nonconforming tetrahedral finite elements for the biharmonic equation

In this article, a family of $H^{2$-nonconforming} finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3D. In the family, the $P\ell$ polynomial space is enriched by some high order polynomials for all $\ell\ge 3$ and the corresponding finite element solution converges at the optimal order $\ell-1$ in $H^2$ norm. Moreover, the result is improved for two low order cases by using $P6$ and $P7$ polynomials to enrich $P4$ and $P_5$ polynomial spaces, respectively. The optimal order error estimate is proved. The numerical results are provided to confirm the theoretical findings.