Augmented saddle point formulation of the steady-state Stefan--Maxwell diffusion problem

TL;DR

This paper develops a structure-preserving finite element method for the steady-state Stefan--Maxwell diffusion problem, ensuring thermodynamic consistency and providing error estimates, with applications to lung gas exchange modeling.

Contribution

It introduces an augmented Lagrangian-inspired approach that yields a symmetric positive definite Onsager matrix and preserves thermodynamic laws in discretization.

Findings

The method guarantees thermodynamic consistency to machine precision.

Error estimates are established for arbitrary species numbers.

Numerical examples demonstrate the method's effectiveness, including lung diffusion modeling.

Abstract

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.