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

TL;DR

This paper introduces a new family of $H^2$-nonconforming finite elements on tetrahedral grids for 3D biharmonic equations, achieving optimal convergence rates and verified by numerical experiments.

Contribution

The paper develops a novel family of $H^2$-nonconforming finite elements for 3D biharmonic problems with proven optimal convergence and improved low-order cases.

Findings

Finite element solutions converge at order $ ext{l}-1$ in $H^2$ norm.

Enrichment with high order polynomials improves accuracy.

Numerical results confirm theoretical error estimates.

Abstract

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 $P_6$ and $P_7$ polynomials to enrich $P_4$ and $P_5$ polynomial spaces, respectively. The optimal order error estimate is proved. The numerical results are provided to confirm the theoretical findings.