The Staggered Discontinuous Galerkin (SDG) method is a class of finite element methods designed to solve partial differential equations while preserving local conservation properties and handling complex geometries. It employs a staggered mesh structure in which scalar and vector variables are discretized on distinct, yet overlapping, primal and dual meshes. This arrangement facilitates a natural enforcement of conservation laws and enables element-wise postprocessing for superconvergent approximations. The SDG framework supports high-order accuracy and geometric flexibility, making it well-suited for problems involving unstructured or polytopal meshes. Moreover, by decoupling the discretization of variables, the method enhances stability and allows for efficient hybridization, yielding compact global systems and connections to other modern finite element approaches such as hybridizable DG, weak Galerkin, and virtual element methods.

In this talk, we present connections between the stabilization-free polygonal element (SF-PE) and staggered discontinuous Galerkin (SDG) methods. By introducing a gradient reconstruction operator, the SDG method can be reformulated as a class of SF-PE methods. Thanks to this connection and some existing SF-PE methods, we first present a new family of polygonal SDG methods utilizing Raviart-Thomas mixed finite element spaces. The inf-sup stability and optimal convergence are proved. Next, with a simple modification of the loading term we are able to obtain globally $\vH(\div)$-conforming velocity fields. We also present a static condensation procedure for the SDG method for its efficient implementation, which can also be applied to the SF-PE methods due to the presented connection. The theoretical results are verified by numerical experiments.