Uniform error analysis of a rectangular Morley finite element method on a Shishkin mesh for a 4th-order singularly perturbed boundary value problem

The singularly perturbed reaction-diffusion problem ${\epsilon }^2{\mathrm{\Delta }}^{2}u-\mathrm{d}\mathrm{i}\mathrm{v}\left(c\mathrm{\nabla }u\right)=f$ is considered on the unit square $\mathrm{\Omega }$ in ${\mathbb{R}}^2$ with homogenous Dirichlet boundary conditions. Its solution typically contains boundary layers on all sides of~$\mathrm{\Omega }$. It is discretised by a finite element method that uses rectangular Morley elements on a Shishkin mesh. In an associated energy-type norm that is natural for this problem, we prove an $O({\epsilon }^{1\mathrm{/}2}{N}^{-1}+\epsilon N^{-1}\ln N+N^{-3\mathrm{/}2})$ rate of convergence for the error in the computed solution, where $N$~is the number of mesh intervals in each coordinate direction. Thus in the most troublesome regime when $\epsilon \approx N^{-1}$, our method is proved to attain an $O(N^{-3\mathrm{/}2})$ rate of convergence, which is shown to be sharp by our numerical experiments and is superior to the $O(N^{-1\mathrm{/}2})$ rate that is proved in Meng & Stynes, Adv. Comput. Math. 2019 when Adini finite elements are used to solve the same problem on the same mesh.