We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}\Delta ^{2}u-\Delta u=f $ on the unit square $\Omega \subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial \Omega$, where $\varepsilon \in (0, 1)$ is a small parameter. The problem is solved numerically by means of a weak Galerkin(WG) finite element method, which is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on finite element partitions consisting of polygons of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with $N^2$ elements is constructed ,convergence of the method is proved in a discrete $H^2$ norm for the corresponding WG finite element solutions and numerical results are presented.

翻译：本文考虑单位正方形$\Omega \subset \mathbb{R}^2$上奇异摄动四阶边值问题$\varepsilon ^{2}\Delta ^{2}u-\Delta u=f $，边界条件为$\partial \Omega$上$u = \partial u / \partial n = 0$，其中$\varepsilon \in (0, 1)$是小参数。采用弱Galerkin（WG）有限元方法进行数值求解，该方法通过使用有限元剖分上任意形状多边形的间断分片多项式，在单元构造中具有高度鲁棒性和灵活性。所得的WG有限元格式对称、正定且无参数。在对解中出现的边界层结构进行合理假设下，构造了含$N^2$个单元的合适Shishkin网格族，证明了相应WG有限元解在离散$H^2$范数下的收敛性，并给出了数值结果。