Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

TL;DR

This paper analyzes the convergence rates of the rectangular Morley element scheme for second order problems across arbitrary dimensions, establishing theoretical error bounds and confirming them with numerical examples.

Contribution

It provides the first comprehensive convergence analysis of the rectangular Morley element scheme in arbitrary dimensions, including error estimates and numerical validation.

Findings

Convergence order of $\\mathcal{O}(h)$ in energy norm for general grids

Convergence order of $\\mathcal{O}(h^2)$ in $L^2$ norm for general grids

Enhanced convergence rates of $\\mathcal{O}(h^2)$ in energy norm on uniform grids

Abstract

In this paper, we present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of $\mathcal{O}(h)$ order in energy norm and of $\mathcal{O}(h^2)$ order in $L^2$ norm on general $d$-rectangular grids. Moreover, when the grid is uniform, the convergence rate can be of $\mathcal{O}(h^2)$ order in energy norm, and the convergence rate in $L^2$ norm is still of $\mathcal{O}(h^2)$ order, which can not be improved. Numerical examples are presented to demonstrate our theoretical results.