  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.
