For the anisotropic problems in 2-D unbounded domains, this paper proposes a natural BEM with non-uniform grids on the elliptic boundary and a coupling of NBEM and FEM with non-uniform grids on the elliptic artificial boundary. Then, the convergence theorems of the NBEM and the coupling method are proved. In addition, the moving mesh method based on equidistribution principle is introduced into the NBEM and the coupling method, respectively. Numerical examples verify the convergence theorems and demonstrate the advantage in accuracy and efficiency of the proposed methods.