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.

针对四阶椭圆奇异摄动问题，提出了一种具有简单修改过的右侧的Morley-Wang-Xu（MWX）单元方法，其中离散双线性形式是通常的非协调有限元方法的标准方法。对此MWX元素方法给出了尖锐的误差分析。而且，在边界层的情况下，将Nitsche的技术应用于MXW元素方法以实现最佳收敛速度。MWX元素方法的一个重要功能是易于求解。基于二维的离散Stokes复数，将MWX元素方法解耦为一个Poisson方程的Lagrange元素方法，两个Poisson方程的Morley元素方法和一个不合格\（P_1 \） – \（P_0 \）Brinkman问题的有限元方法，这意味着MWX元素方法的有效且鲁棒的求解器。提供了一些数值例子来验证理论结果。