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

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 ℓ polynomial space is enriched by some high order polynomials for all ℓ ≥ 3 and the corresponding finite element solution converges at the optimal order ℓ − 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 P5 polynomial spaces, respectively. The optimal order error estimate is proved. The numerical results are provided to confirm the theoretical findings.2000 Mathematics Subject Classification. 65N15, 65N30S. Key words and phrases. nonconforming H 2 element, finite element method, biharmonic problem, tetrahedral grid.