In order to improve the effectiveness and stability of the coarse-mesh finite difference method (CMFD), we developed a new nonlinear diffusion acceleration scheme for solving neutron transport equations. This scheme, called LR-NDA, employs a local refinement approach on the framework of CMFD by solving a local boundary value problem of the scalar flux on the coarse-mesh structure to replace the piecewise constant scalar flux obtained by CMFD. The refined flux is then used to update the scalar flux in the neutron transport source iteration. In this paper, a detailed convergence study of LR-NDA is carried out based on a two-dimensional fixed-source problem, and it shows that LR-NDA is much more effective and stable than CMFD for a wide range of optical thicknesses. In addition, we demonstrate that LR-NDA is a local adaptive method. LR-NDA does not necessarily require local refinement for all the coarse-mesh cells on the problem domain, i.e., it can be used only for relatively optically thick regions where the standard CMFD scheme would encounter the convergence problem.