In this article, a staggered discontinuous Galerkin (SDG) approximation on rectangular meshes for elliptic problems in two dimensions is constructed and analyzed. The optimal convergence results with respect to discrete $L^2$ and $H^1$ norms are theoretically proved. Some numerical evidences to verify the optimal convergence rates are presented. Several numerical examples to the elliptic singularly perturbed problems with sharp boundary or interior layers are presented to show that the proposed SDG method is very effective, stable and accurate. Thanks to the simple structure of rectangular meshes, the discrete gradients across the boundaries of rectangular elements are easily defined, making numerical implementation much easier. The idea of using the rectangular meshes will be extended to more practical problems on a curved domain in future works.

在本文中，构造并分析了二维椭圆问题的矩形网格上的交错不连续伽辽金 (SDG) 近似。关于离散的最优收敛结果${升}^2$ 和 ${高}^1$规范得到了理论上的证明。给出了一些验证最优收敛速度的数值证据。给出了具有尖锐边界或内层的椭圆奇异摄动问题的几个数值例子，表明所提出的 SDG 方法非常有效、稳定和准确。由于矩形网格的简单结构，跨越矩形元素边界的离散梯度很容易定义，使得数值实现更加容易。在未来的工作中，使用矩形网格的想法将扩展到弯曲域上更实际的问题。