Variational aspects of the reaction in the method of moments

The reaction R(J₁, J₂) between two electric surface current densities J₁ and J₂ is the integral of the dot product of J₁ with the electric field produced by J₂. Jack H. Richmond (see ibid., vol.39, no.4, p.473-479, 1991) has correctly shown that the reaction is variational (a formula is variational if it has a stationary point at the quantity of interest) with Galerkin´s method but is not necessarily variational when the non-Galerkin element method is applied in the manner where one set of functions is used to expand both J₁ and J₂ and another set of functions is used to test both the integral equation for J₁ and that for J₂. In this paper, the reaction R(J₁, J₂) is shown to be variational when the non-Galerkin method is applied in the manner of operation where the set of functions used to expand J₂ is the set of functions used to test the integral equation for J₁ and the set of functions used to test the integral equation for J₂ is the set of functions used to expand J₁. Hence, the manner of operation just described may be more appropriate than Richmond´s if a variational result is desired. A few numerical results are included to compare the performance of both manners of operation