This is the first paper of a series in which the author presents a new mixed formulation for the numerical solution of second-order elliptic problems. This new formulation expands the standard mixed formulation in the sense that three variables are explicitly treated: the scalar unknown, its gradient, and its flux (the coefficient times the gradient). Based on this formulation, mixed finite element approximations of the second-order elliptic problems are considered. Optimal order error estimates in the

${L}^{p}$- and

${H}^{-s}$-norms are obtained for the mixed approximations. Various implementation techniques for solving systems of algebraic equations are discussed. A postprocessing method for improving the scalar variable is analyzed, and superconvergent estimates in the

${L}^{p}$-norm are derived. The mixed formulation is suitable for the case where the coefficient of differential equations is a small tensor and does not need to be inverted.