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 $\varepsilon^2\Delta^2 u - \mathrm{div}\left(c\nabla u\right) = f$ is considered on the unit square $\Omega$ in $\mathbb{R}^2$ with homogenous Dirichlet boundary conditions. Its solution typically contains boundary layers on all sides of~$\Omega$. 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 $O(\varepsilon^{1/2}N^{-1}+\varepsilon N^{-1}\ln N + N^{-3/2})$ 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 $\varepsilon \approx N^{-1}$, our method is proved to attain an $O(N^{-3/2})$ rate of convergence, which is shown to be sharp by our numerical experiments and is superior to the $O(N^{-1/2})$ 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.