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This is the first in a series of three papers in which the authors discuss a method for post-processing a finite element solution to obtain high accuracy approximations for displacements, stresses, stress intensity factors etc. Rather than take the values of these quantities directly from the finite element solution, they evaluate certain weighted averages of the solution over the entire region. These yield approximations that are of the same order of accuracy as the strain energy. They obtain error estimates, and also present some numerical examples to illustrate the practical effectively of the technique. In the third paper of this series the authors address the matters of adaptive mesh selection and a'posteriori error estimation. (Author)