Steady-state and time-dependent currents under diffusion limited conditions for a circular microelectrode embedded in insulating material are computed using the finite element method. A locally valid series expansion is used to correct for the boundary singularity at the electrode edge. By incorporating this correction into the equations of the finite element method in a fully implicit manner, we are able to obtain values of the time-dependent current with better than 1% accuracy on meshes (typically about 400 points) with approximately two orders of magnitude fewer points than previously reported.