A staggered discontinuous Galerkin method for elliptic problems on rectangular grids

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 L² and H¹ 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.