  A Morley-Wang-Xu (MWX) element method with a simply modified right hand side
is proposed for a fourth order elliptic singular perturbation problem, in which
the discrete bilinear form is standard as usual nonconforming finite element
methods. The sharp error analysis is given for this MWX element method. And the
Nitsche's technique is applied to the MXW element method to achieve the optimal
convergence rate in the case of the boundary layers. An important feature of
the MWX element method is solver-friendly. Based on a discrete Stokes complex
in two dimensions, the MWX element method is decoupled into one Lagrange
element method of Poisson equation, two Morley element methods of Poisson
equation and one nonconforming $P_1$-$P_0$ element method of Brinkman problem,
which implies efficient and robust solvers for the MWX element method. Some
numerical examples are provided to verify the theoretical results.
