It is known that a standard uniform mesh refinement of a three-dimensional polyhedral domain loses effectiveness of the finite element method, which leads to the slow convergence rate of numerical solution depending on the geometric structure and the imposed boundary conditions. Some existing studies show that a non-uniform graded mesh refinement using tetrahedron elements can overcome such an issue. The technique handles singular edges and singular corners due to the geometric structure and boundary conditions, and improve the convergence rate of the numerical solution of the finite element method. This work is to generalize the existing method not only tetrahedron elements but also other typical ones such as wedge elements and hexahedron elements. Both theoretical study and results of numerical experiments of this work will be presented.