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Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. An error estimate in H¹ norm is obtained for a piecewise finite element approximation to the solution of the nonlinear steady-state Poisson-Nernst-Planck equations. Some superconvergence results are also derived by using the gradient recovery technique for the equations. Numerical results are given to validate the theoretical results. It is also numerically illustrated that the gradient recovery technique can be successfully applied to the computation of the practical ion channel problem to improve the efficiency of the external iteration and save CPU time.