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

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.