LNG_FEM: Graded meshes on domains of polygonal structures We develop LNG_FEM, a software package for graded mesh generation and for solving elliptic equations. LNG_FEM generates user-specified graded meshes on arbitrary 2D domains with straight edges for different boundary conditions. We focus on a detailed exposition on the implementation of the software. In addition, we demonstrate that LNG_FEM is equipped with advanced algorithms and data structures to perform efficiently in numerical tests. We hope that LNG_FEM can broaden the use and understanding of graded meshes in the finite element approximation of singular solutions.