After a brief description of GYARMATI's Governing Principle of Dissipative Processes, general approximation methods are developed for heat conduction phenomena on two scales. On the first scale the temperature and heat current density fields are approximated by two different sets of functions in such a way that the internal energy balance and the imposed boundary conditions be satisfied. On the second scale we introduce a new temperature field related to the heat current density through the constitutive equation. Since the dissipation potentials in GYARMATI's principle are connected with each other by LEGENDRE dual transformations we call “dual field methods” the approximation procedure based on the two temperature fields. It is shown that the equations of the GALERKIN method, the method of orthogonal projections, and the TREFFTZ method, when applied to the appropriate problems, are included in the approximation schemes. Due to their great importance a special paragraph is devoted to the treatment of eigenvalue problems. Finally, the approximation methods are illustrated by a simple example and the results are compared with the exact solution.