  We investigate structure-preserving finite element discretizations of the
steady-state Stefan--Maxwell diffusion problem which governs diffusion within a
phase consisting of multiple species. An approach inspired by augmented
Lagrangian methods allows us to construct a symmetric positive definite
augmented Onsager transport matrix, which in turn leads to an effective
numerical algorithm. We prove inf-sup conditions for the continuous and
discrete linearized systems and obtain error estimates for a phase consisting
of an arbitrary number of species. The discretization preserves the
thermodynamically fundamental Gibbs--Duhem equation to machine precision
independent of mesh size. The results are illustrated with numerical examples,
including an application to modelling the diffusion of oxygen, carbon dioxide,
water vapour and nitrogen in the lungs.
