The Adini finite element on locally refined meshes

TL;DR

This paper develops a locally refined Adini finite element method for the biharmonic equation, analyzing its convergence properties and proposing an effective residual-based error estimator.

Contribution

It introduces a locally refined Adini finite element scheme, analyzes convergence behavior under different normal derivative assignments, and validates an explicit error estimator.

Findings

Superlinear convergence ($h^{3/2}$) achieved with averaging normal derivatives on uniform meshes.

Convergence order $h^2$ is lost with certain derivative assignments on refined meshes.

The residual-based error estimator is reliable and efficient.

Abstract

This work introduces a locally refined version of the Adini finite element for the planar biharmonic equation on rectangular partitions with at most one hanging node per edge. If global continuity of the discrete functions is enforced, for such method there is some freedom in assigning the normal derivative degree of freedom at the hanging nodes. It is proven that the convergence order $h^2$ known for regular solutions and regular partitions is lost for any such choice, and that assigning the average of the normal derivatives at the neighbouring regular vertices is the only choice that achieves a superlinear order, namely $h^{3/2}$ on uniformly refined meshes. On adaptive meshes, the method behaves like a first-order scheme. Furthermore, the reliability and efficiency of an explicit residual-based error estimator are shown up to the best approximation of the Hessian by certain piecewise polynomial functions.