In this article we are concerned with the discontinuous Galerkin (DG) method in connection with the numerical solution of the state problem in the field of shape optimization techniques. The presented state problem is described by the stationary energy equation of the system of the mould, glass piece, plunger and plunger cavity arising from the forming process in the glass industry. The attention is paid to the development of the numerical scheme based on the piecewise polynomial, generally discontinuous approximation, which enables to better resolve various phenomena typical for such a heterogeneous medium problem, compared with standard common numerical techniques. The studied problem is supplemented with the preliminary numerical results demonstrating the potency of the proposed scheme.