  The porous medium equation (PME) is a typical nonlinear degenerate parabolic
equation. We have studied numerical methods for PME by an energetic variational
approach in [C. Duan et al, J. Comput. Phys., 385 (2019) 13-32], where the
trajectory equation can be obtained and two numerical schemes have been
developed based on different dissipative energy laws. It is also proved that
the nonlinear scheme, based on $f \log f$ as the total energy form of the
dissipative law, is uniquely solvable on an admissible convex set and preserves
the corresponding discrete dissipation law. Moreover, under certain smoothness
assumption, we have also obtained the second order convergence in space and the
first order convergence in time for the scheme. In this paper, we provide a
rigorous proof of the error estimate by a careful higher order asymptotic
expansion and two step error estimates. The latter technique contains a rough
estimate to control the highly nonlinear term in a discrete $W^{1,\infty}$
norm, and a refined estimate is applied to derive the optimal error order.
