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A nonconforming finite element method (FEM) is developed and investigated for the coupled Poisson–Nernst–Planck (PNP) equations with low order EQ1rot element. Then, by use of the special properties of this element, that is, the interpolation operator is equivalent to its projection operator, and the consistency error estimate can reach order of O(h2) which is one order higher than that of its interpolation error estimates when the exact solution belongs to H3(Ω), the superclose estimates of order O(h2) and O(h2 + τ) in the broken H1‐norm are derived with new techniques for the semidiscrete scheme and backward Euler fully discrete scheme, respectively. Further, through employing interpolation postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some numerical results are provided to confirm the theoretical analysis. It seems that our results have never been found in the existing literature. Here h and τ denote the mesh size and time step, respectively. [ABSTRACT FROM AUTHOR]