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Algebraic methods, like the cell method or the finite integration technique are known to be effective in solving numerical problems, but they are limited to linear convergence, i.e., they exactly reconstruct constant fields inside the element. A few attempts in the literature have been aimed at extending the method to higher order, but results have not been completely satisfactory. This paper proposes a novel technique to extend the cell method to second order convergence. The consistency and convergence of the proposed approach are established by numerical results.