The boundary element method (BEM) has been applied to solve numerically the equations describing steady state convective electrodiffusion through homogeneous and inhomogeneous charged membranes. The numerical procedure is based on that employed by Ramachandran [J. Comput. Phys. 102, 63 (1992)] for diffusion-reaction problems, and directly provides the concentration and concentration gradient profiles through the membrane. From these, the electric profile and the ion fluxes are readily obtained. The numerical solutions are compared with the analytical ones whenever possible. It is found that the BEM constitutes a convenient alternative to the finite difference techniques previously developed for solving the Nernst-Planck equations of membrane electrochemistry in those cases where space charge effects can be ignored.