Recently, the Local Discontinuous Galerkin (LDG) method has been successfully applied to solve various reaction-diffusion problems characterized by boundary layers. To achieve uniform convergence in the small perturbation parameter \(\varepsilon \) , layer-adapted meshes of Shishkin-type along with their variants, have been frequently combined with the numerical schemes. However, limited convergence results have been reported for the LDG method applied to graded meshes. These meshes, generated through recursive formulae, present a compelling alternative, especially they are less sensitive to variations in \(\varepsilon \) , which determines the mesh structure and the solution behavior. In the present paper, we investigate the convergence of the LDG method on two Duran-type meshes: the Duran-Shishkin mesh and the Duran mesh. We derive optimal-order error estimates if the logarithmic factor is neglected. These results, appearing in the literature for the first time, open up new possibilities for the application of the LDG method with graded meshes in solving problems with boundary layers. Numerical results confirm that our theoretical estimates are sharp.
