  In this paper, we present and analyze a staggered discontinuous Galerkin
method for Darcy flows in fractured porous media on fairly general meshes. A
staggered discontinuous Galerkin method and a standard conforming finite
element method with appropriate inclusion of interface conditions are exploited
for the bulk region and the fracture, respectively. Our current analysis
weakens the usual assumption on the polygonal mesh, which can integrate more
general meshes such as elements with arbitrarily small edges into our
theoretical framework. We prove the optimal convergence estimates in $L^2$
error for all the variables by exploiting the Ritz projection. Importantly, our
error estimates are shown to be fully robust with respect to the heterogeneity
and anisotropy of the permeability coefficients. Several numerical experiments
including meshes with small edges and anisotropic meshes are carried out to
confirm the theoretical findings. Finally, our method is applied in the
framework of unfitted mesh.
