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The finite-element method is widely used for the solution of field problems but the method, as generally applied, suffers from the fact that it is not known how close the solution is to the actual value. This uncertainty can be reduced by providing a dual finite-element method which is so arranged that both methods together provide upper and lower bounds to the correct solution. It is found that the double approach also promises economies in the computation. The paper examines the physical basis of the dual method and applies it to Laplacian and Poissonian problems.