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 ε²Δ² u - div(c∇ u) = f is considered on the unit square $Ω$ in R² with homogenous Dirichlet boundary conditions. Its solution typically contains boundary layers on all sides of~$Ω$. 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(ε1/2N⁻¹+ε N⁻¹ln 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 ε ≈ 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.