The article presents a mathematical apparatus for solving the radiation problems by the nodal finite elements method in a two-dimensional field. We obtain the final expression for the elements of the local matrix and the methodology for deriving the functional dependence of the excitation field of a point source (for forming the right side of the vector). The calculation test performed by this method improved the model of elementary electric dipole in the form of a flat frame with electric current. The reliability of the results confirmed a high coincidence with the analytical solution of the radiation problem of the elementary electric dipole. The results of the article can be used in the calculations of any axially symmetric antenna excited by a point source, including the problems of structural synthesis of axially symmetric emitters for different purposes.