This paper consists of a consistent codification and generalization of previous formalisms and of extensions that result in new approximation methods. The discussion is restricted to the multigroup neutron diffusion equations. The development of variational principles that admit discontinuous trial functions which need satisfy neither the final and initial conditions nor the external boundary conditions of the physical problem is reviewed and generalized. Consistent single-channel and multichannel spatial synthesis and spectral synthesis formalisms are developed. The difficulty with over determined interface conditions which has arisen in previous work is traced to a failure to properly apply continuity requirements at these interfaces so as to relate variations in the adjoint trial functions on opposite sides of the interface. A generalized nodal formalism is developed as an extension of the variational synthesis method. The use of a particular type of piecewise cubic polynomial to modulate expansion functions is introduced as a means to obtain overlapping multichannel synthesis and generalized nodal approximations which do not require the evaluation of troublesome surface integrals. A brief review of the experience to date with variational synthesis methods for the multigroup neutron diffusion equations is given.
