Rectangular $C^1$-$Q_k$ Bell Finite Elements in Two and Three Dimensions

Abstract

For a $Q_k$ polynomial, its normal derivatives on the element boundaries are still $P_k$ ($Q_k$ in three dimensions) polynomials. For a Bell $Q_k$ finite element function, its normal derivatives on the element boundaries are $P_{k-1}$ ($Q_{k-1}$ in three dimensions) polynomials. We construct a Bell $C^1$-$Q_k$ finite element on rectangular meshes in two dimensions and three dimensions for $k\geqslant 4$. We show, with a big reduction from the standard Bogner-Fox-Schmit (BFS) $C^1$-$Q_k$ finite element, the $C^1$-$Q_k$ Bell finite element retains the optimal order of convergence. Numerical experiments are performed, comparing the new elements with the original elements.

Keywords

Biharmonic equation / Conforming element / Rectangular element / Finite element / Rectangular mesh / Cuboid mesh / 65N15 / 65N30

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Funding

National Natural Science Foundation of China(12071128)

RIGHTS & PERMISSIONS

Shanghai University