On the Use of Dual Variational Principles for the Estimation of Error in Approximate Solutions of Diffusion Problems

An exploration is made into a method for using reciprocal variational problems to develop figures of merit for approximate solutions of diffusion problems. The theory of the reciprocal problems is described in both a continuous and discrete context. Connections with the method of Slobodyansky are discussed. A strategem is presented for extending the method to the (non-self-adjoint) group-diffusion case. Limitations of the method are discussed and numerical examples given. It is concluded that the method is useful in one-, two-, and perhaps in small three- dimensional problems but is probably computationally not practical for full-blown, detailed, three-dimensional calculations.