Capacity of the Adini element for biharmonic equations

TL;DR

This paper analyzes the convergence properties of the Adini element scheme for fourth order problems, establishing optimal convergence rates in energy and $L^2$ norms with numerical validation.

Contribution

It provides the first rigorous proof of the convergence order of the Adini element scheme for biharmonic equations in arbitrary dimensions.

Findings

Convergence order in energy norm is $igo(h^2)$ for solutions in $H^4$.

Convergence rate in $L^2$ norm cannot exceed $igo(h^2)$.

Numerical experiments confirm theoretical results.

Abstract

This paper is devoted to the convergence analysis of the Adini element scheme for the fourth order problem in arbitrary dimension. We prove that, the Adini element scheme is $\mathcal{O}(h^2)$ order convergent in energy norm provided the exact solution is in $H^4$, and the convergence rate in $L^2$ norm can not be nontrivially higher than $\mathcal{O}(h^2)$ order. Numerical verifications are presented.