  In this paper, we will provide the the finite element method for the
electro-osmotic flow in micro-channels, in which a convection-diffusion type
equation is given for the charge density $\rho^e$. A time-discrete method based
on the backward Euler method is designed. The theoretical analysis shows that
the numerical algorithm is unconditionally stable and has optimal convergence
rates. To show the effectiveness of the proposed model, some numerical results
for the electro-osmotic flow in the T-junction micro-channels and in rough
micro-channels are provided. Numerical results indicate that the proposed
numerical method is suitable for simulating electro-osmotic flows.
