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This paper demonstrates finite element procedure for two-dimensional axisymmetric domains. For many engineering applications like structural engineering, aerospace engineering, geo-mechanics etc., the solution domain and boundary conditions are axisymmetric. Henceforth, we can illuminate just the axisymmetric part of the solution domain that gives the data of the entire domain. This paper demonstrates the effectiveness of using MATLAB programming demonstrated by Persson et.al (2004) as the initial mesh for discretization of axisymmetric domains for higher order meshing. Further, solving some class of partial differential equations using finite element method with nodal relation given by subparametric transformations Rathod et.al (2008). In this paper a cubic order curved triangular meshing for some of the domains like ellipse and circle are demonstrated. These in turn finds its applications in the fields like stress analysis in mechanical engineering, torsion twist (shear strength) analysis in civil engineering, evaluation of stress intensity factor for quarter elliptical crack in pressure vessels in equipment industry etc,. The output data from the meshing scheme like meshing of the domain, nodal position, element connectivity and boundary edges are been used in the finite element procedures. The efficiency of the method is achieved by p-refinement scheme i.e., fixing the number of elements and increasing the polynomial order.